1. Giới thiệu về khái niệm liệt kê và tầm quan trọng trong toán học Trong chương trình toán học lớp 10, "liệt kê" là một trong những công cụ đắc lực giúp học sinh giải quyết các bài toán liên quan đến đếm, xác suất, tổ hợp,... Liệt kê cho phép ta tìm ra tất cả những khả năng hoặc phương án có thể xảy ra mà không bỏ sót bất kỳ trường hợp nào. Đây là một bước quan trọng khi giải các bài toán về tập hợp, xác suất hoặc khi cần kiểm tra các trường hợp đặc biệt trong quá trình chứng minh. Học sinh nắm vững kỹ năng liệt kê sẽ dễ dàng tiếp cận và giải quyết các bài toán tổ hợp, xác suất, thống kê và nhiều lĩnh vực toán học khác.
2. Định nghĩa khái niệm liệt kê Liệt kê trong toán học là phương pháp tìm và viết ra toàn bộ các phần tử của một tập hợp dựa trên một quy tắc xác định, sao cho không bỏ sót hoặc lặp lại bất kỳ phần tử nào.
Nói một cách đơn giản, liệt kê là quá trình ghi lại tất cả các khả năng hoặc phần tử thỏa mãn điều kiện đề bài.
3. Các bước liệt kê và ví dụ minh họa Quy trình liệt kê thường bao gồm các bước như sau:
1. Xác định rõ điều kiện hoặc quy tắc để lựa chọn phần tử. 2. Sử dụng phương pháp hệ thống (theo thứ tự xác định) để tránh bỏ sót hoặc lặp lại. 3. Kiểm tra lại kết quả sau khi liệt kê để đảm bảo đầy đủ, không trùng lặp. Ví dụ 1: Liệt kê tập hợp các số tự nhiên nhỏ hơn 5. Các số tự nhiên nhỏ hơn 5 là 0, 1, 2, 3, 4. Vậy tập hợp đó là:{ 0 ; 1 ; 2 ; 3 ; 4 } \{0; 1; 2; 3; 4\} { 0 ; 1 ; 2 ; 3 ; 4 }
Ví dụ 2: Liệt kê các số nguyên có hai chữ số mà tổng các chữ số bằng 4. Gọi số cần tìm là a b ab ab a a a b b b a ≥ 1 a \geq 1 a ≥ 1 b ≥ 0 b \geq 0 b ≥ 0 a + b = 4 a + b = 4 a + b = 4
a = 1 , b = 3 → 13 a = 1, b = 3 \rightarrow 13 a = 1 , b = 3 → 13 a = 2 , b = 2 → 22 a = 2, b = 2 \rightarrow 22 a = 2 , b = 2 → 22 a = 3 , b = 1 → 31 a = 3, b = 1 \rightarrow 31 a = 3 , b = 1 → 31 a = 4 , b = 0 → 40 a = 4, b = 0 \rightarrow 40 a = 4 , b = 0 → 40 Vậy các số cần liệt kê là:13 13 13 22 22 22 31 31 31 40 40 40
4. Các trường hợp đặc biệt và lưu ý khi liệt kê Cần phân tích điều kiện bài toán thật rõ ràng, tránh nhầm lẫn giữa các trường hợp đặc biệt (ví dụ: số có thể lặp chữ số hay không, có thể là số âm hay không, v.v.). Nên cố gắng liệt kê theo một chuỗi logic (tăng dần, giảm dần, theo nhóm, ...) để không bỏ sót. Với các bài toán phức tạp, nên vẽ bảng hoặc biểu đồ để kiểm soát các trường hợp dễ hơn. 5. Mối liên hệ của liệt kê với các khái niệm toán học khác Liệt kê giúp xác định số phần tử của một tập hợp – là nền tảng cho các công thức tổ hợp như chỉnh hợp, hoán vị, tổ hợp. Trong xác suất, liệt kê tất cả biến cố có thể xảy ra giúp ta tính toán xác suất đúng và đầy đủ. Liệt kê thường kết hợp với sử dụng cây liệt kê (sơ đồ cây) để hệ thống tất cả trường hợp có thể xảy ra. 6. Bài tập mẫu có lời giải chi tiết Bài tập 1: Liệt kê tất cả các số có ba chữ số mà tổng các chữ số bằng 6. Đặt số cần tìm là a b c abc ab c a , b , c a, b, c a , b , c a ≥ 1 a \geq 1 a ≥ 1 ( a , b , c ) (a, b, c) ( a , b , c ) a + b + c = 6 a + b + c = 6 a + b + c = 6 a ∈ { 1 , 2 , . . . , 6 } a \in \{1, 2,..., 6\} a ∈ { 1 , 2 , ... , 6 } b , c ∈ { 0 , 1 , . . . , 9 } b, c \in \{0, 1,..., 9\} b , c ∈ { 0 , 1 , ... , 9 }
Ta xét các trường hợp theoa a a
a = 1 a=1 a = 1 b + c = 5 b + c = 5 b + c = 5 ( b , c ) = ( 0 , 5 ) ; ( 1 , 4 ) ; ( 2 , 3 ) ; ( 3 , 2 ) ; ( 4 , 1 ) ; ( 5 , 0 ) → (b, c) = (0,5); (1,4); (2,3); (3,2); (4,1); (5,0) \rightarrow ( b , c ) = ( 0 , 5 ) ; ( 1 , 4 ) ; ( 2 , 3 ) ; ( 3 , 2 ) ; ( 4 , 1 ) ; ( 5 , 0 ) → a = 2 a=2 a = 2 b + c = 4 b + c = 4 b + c = 4 ( b , c ) = ( 0 , 4 ) ; ( 1 , 3 ) ; ( 2 , 2 ) ; ( 3 , 1 ) ; ( 4 , 0 ) → (b, c) = (0,4); (1,3); (2,2); (3,1); (4,0) \rightarrow ( b , c ) = ( 0 , 4 ) ; ( 1 , 3 ) ; ( 2 , 2 ) ; ( 3 , 1 ) ; ( 4 , 0 ) → < ! − − L A T E X P R O C E S S E D 1 754127424270 − − > < / l i > < l i > < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > a < / m i > < m o > = < / m o > < m n > 2 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > a = 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > : < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > b < / m i > < m o > + < / m o > < m i > c < / m i > < m o > = < / m o > < m n > 4 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > b + c = 4 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7778 e m ; v e r t i c a l − a l i g n : − 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > c < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < b r > − < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > b < / m i > < m o s e p a r a t o r = " t r u e " > , < / m o > < m i > c < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 4 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s e p a r a t o r = " t r u e " > ; < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 1 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 3 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s e p a r a t o r = " t r u e " > ; < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 2 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 2 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s e p a r a t o r = " t r u e " > ; < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 3 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 1 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s e p a r a t o r = " t r u e " > ; < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 4 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 0 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > → < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > ( b , c ) = ( 0 , 4 ) ; ( 1 , 3 ) ; ( 2 , 2 ) ; ( 3 , 1 ) ; ( 4 , 0 ) → < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > c < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m p u n c t " > ; < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m p u n c t " > ; < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m p u n c t " > ; < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m p u n c t " > ; < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > → < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > 204 , 213 , 222 , 231 , 240 <!--LATEX_PROCESSED_1754127424270--></li><li><span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">a=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span>:<span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">b + c = 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span><br>-<span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">)</mo><mo separator="true">;</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo separator="true">;</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">;</mo><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">;</mo><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo></mrow><annotation encoding="application/x-tex">(b, c) = (0,4); (1,3); (2,2); (3,1); (4,0) \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span></span>204, 213, 222, 231, 240 < ! − − L A TE X P ROCESSE D 1 754127424270 − − >< / l i >< l i >< s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > a < / mi >< m o >=< / m o >< mn > 2 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > a = 2 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >:< s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > b < / mi >< m o > + < / m o >< mi > c < / mi >< m o >=< / m o >< mn > 4 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > b + c = 4 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7778 e m ; v er t i c a l − a l i g n : − 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > c < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 4 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< b r > − < s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< mi > b < / mi >< m ose p a r a t or = " t r u e " > , < / m o >< mi > c < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 4 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m ose p a r a t or = " t r u e " > ; < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 1 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 3 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m ose p a r a t or = " t r u e " > ; < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 2 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 2 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m ose p a r a t or = " t r u e " > ; < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 3 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 1 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m ose p a r a t or = " t r u e " > ; < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 4 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 0 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >→< / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > ( b , c ) = ( 0 , 4 ) ; ( 1 , 3 ) ; ( 2 , 2 ) ; ( 3 , 1 ) ; ( 4 , 0 ) →< / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > c < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m p u n c t " > ; < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m p u n c t " > ; < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m p u n c t " > ; < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 3 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m p u n c t " > ; < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >→< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > 204 , 213 , 222 , 231 , 240 $ a = 2 a=2 a = 2 b + c = 4 b + c = 4 b + c = 4 ( b , c ) = ( 0 , 4 ) ; ( 1 , 3 ) ; ( 2 , 2 ) ; ( 3 , 1 ) ; ( 4 , 0 ) → (b, c) = (0,4); (1,3); (2,2); (3,1); (4,0) \rightarrow ( b , c ) = ( 0 , 4 ) ; ( 1 , 3 ) ; ( 2 , 2 ) ; ( 3 , 1 ) ; ( 4 , 0 ) → a = 3 a=3 a = 3 b + c = 3 b + c = 3 b + c = 3 ( b , c ) = ( 0 , 3 ) ; ( 1 , 2 ) ; ( 2 , 1 ) ; ( 3 , 0 ) → (b, c) = (0,3); (1,2); (2,1); (3,0) \rightarrow ( b , c ) = ( 0 , 3 ) ; ( 1 , 2 ) ; ( 2 , 1 ) ; ( 3 , 0 ) → a = 4 a=4 a = 4 b + c = 2 b + c = 2 b + c = 2 ( b , c ) = ( 0 , 2 ) ; ( 1 , 1 ) ; ( 2 , 0 ) → (b, c) = (0,2); (1,1); (2,0) \rightarrow ( b , c ) = ( 0 , 2 ) ; ( 1 , 1 ) ; ( 2 , 0 ) → < ! − − L A T E X P R O C E S S E D 1 754127424272 − − > < / l i > < l i > < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > a < / m i > < m o > = < / m o > < m n > 4 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > a = 4 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > : < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > b < / m i > < m o > + < / m o > < m i > c < / m i > < m o > = < / m o > < m n > 2 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > b + c = 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7778 e m ; v e r t i c a l − a l i g n : − 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > c < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < b r > − < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > b < / m i > < m o s e p a r a t o r = " t r u e " > , < / m o > < m i > c < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 2 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s e p a r a t o r = " t r u e " > ; < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 1 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 1 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o s e p a r a t o r = " t r u e " > ; < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 2 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 0 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > → < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > ( b , c ) = ( 0 , 2 ) ; ( 1 , 1 ) ; ( 2 , 0 ) → < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > c < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m p u n c t " > ; < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m p u n c t " > ; < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > → < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > 402 , 411 , 420 <!--LATEX_PROCESSED_1754127424272--></li><li><span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">a=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span>:<span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">b + c = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span><br>-<span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">;</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">;</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo></mrow><annotation encoding="application/x-tex">(b, c) = (0,2); (1,1); (2,0) \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span></span>402, 411, 420 < ! − − L A TE X P ROCESSE D 1 754127424272 − − >< / l i >< l i >< s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > a < / mi >< m o >=< / m o >< mn > 4 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > a = 4 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 4 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >:< s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > b < / mi >< m o > + < / m o >< mi > c < / mi >< m o >=< / m o >< mn > 2 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > b + c = 2 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7778 e m ; v er t i c a l − a l i g n : − 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > c < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< b r > − < s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< mi > b < / mi >< m ose p a r a t or = " t r u e " > , < / m o >< mi > c < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 2 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m ose p a r a t or = " t r u e " > ; < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 1 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 1 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m ose p a r a t or = " t r u e " > ; < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 2 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 0 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >→< / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > ( b , c ) = ( 0 , 2 ) ; ( 1 , 1 ) ; ( 2 , 0 ) →< / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > c < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m p u n c t " > ; < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m p u n c t " > ; < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >→< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > 402 , 411 , 420 $ a = 4 a=4 a = 4 b + c = 2 b + c = 2 b + c = 2 ( b , c ) = ( 0 , 2 ) ; ( 1 , 1 ) ; ( 2 , 0 ) → (b, c) = (0,2); (1,1); (2,0) \rightarrow ( b , c ) = ( 0 , 2 ) ; ( 1 , 1 ) ; ( 2 , 0 ) → a = 5 a=5 a = 5 b + c = 1 b + c = 1 b + c = 1 ( b , c ) = ( 0 , 1 ) ; ( 1 , 0 ) → (b, c) = (0,1); (1,0) \rightarrow ( b , c ) = ( 0 , 1 ) ; ( 1 , 0 ) → a = 6 a=6 a = 6 b + c = 0 b + c = 0 b + c = 0 ( b , c ) = ( 0 , 0 ) → (b, c) = (0,0) \rightarrow ( b , c ) = ( 0 , 0 ) → < ! − − L A T E X P R O C E S S E D 1 754127424273 − − > < / l i > < l i > < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > a < / m i > < m o > = < / m o > < m n > 6 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > a = 6 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 6 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > : < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > b < / m i > < m o > + < / m o > < m i > c < / m i > < m o > = < / m o > < m n > 0 < / m n > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > b + c = 0 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7778 e m ; v e r t i c a l − a l i g n : − 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > c < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < b r > − < s p a n c l a s s = " m a t h − i n l i n e " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > b < / m i > < m o s e p a r a t o r = " t r u e " > , < / m o > < m i > c < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m n > 0 < / m n > < m o s e p a r a t o r = " t r u e " > , < / m o > < m n > 0 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > → < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > ( b , c ) = ( 0 , 0 ) → < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > c < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < s p a n c l a s s = " m p u n c t " > , < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 0 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > → < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > 600 <!--LATEX_PROCESSED_1754127424273--></li><li><span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">a=6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span></span>:<span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">b + c = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span><br>-<span class="math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo></mrow><annotation encoding="application/x-tex">(b, c) = (0,0) \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span></span></span></span></span>600 < ! − − L A TE X P ROCESSE D 1 754127424273 − − >< / l i >< l i >< s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > a < / mi >< m o >=< / m o >< mn > 6 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > a = 6 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 6 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >:< s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > b < / mi >< m o > + < / m o >< mi > c < / mi >< m o >=< / m o >< mn > 0 < / mn >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > b + c = 0 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7778 e m ; v er t i c a l − a l i g n : − 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.4306 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > c < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6444 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< b r > − < s p an c l a ss = " ma t h − in l in e " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m os t re t c h y = " f a l se " > ( < / m o >< mi > b < / mi >< m ose p a r a t or = " t r u e " > , < / m o >< mi > c < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m os t re t c h y = " f a l se " > ( < / m o >< mn > 0 < / mn >< m ose p a r a t or = " t r u e " > , < / m o >< mn > 0 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >→< / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > ( b , c ) = ( 0 , 0 ) →< / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > c < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< s p an c l a ss = " m p u n c t " > , < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 0 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >→< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > 600 $ a = 6 a=6 a = 6 b + c = 0 b + c = 0 b + c = 0 ( b , c ) = ( 0 , 0 ) → (b, c) = (0,0) \rightarrow ( b , c ) = ( 0 , 0 ) → Vậy các số cần liệt kê là: < b r / > 105 , 114 , 123 , 132 , 141 , 150 , < b r / > 204 , 213 , 222 , 231 , 240 , < b r / > 303 , 312 , 321 , 330 , < b r / > 402 , 411 , 420 , < b r / > 501 , 510 , < b r / > 600 < b r / > <br />105, 114, 123, 132, 141, 150, <br />204, 213, 222, 231, 240, <br />303, 312, 321, 330, <br />402, 411, 420, <br />501, 510, <br />600<br /> < b r / > 105 , 114 , 123 , 132 , 141 , 150 , < b r / > 204 , 213 , 222 , 231 , 240 , < b r / > 303 , 312 , 321 , 330 , < b r / > 402 , 411 , 420 , < b r / > 501 , 510 , < b r / > 600 < b r / >
Bài tập 2: Liệt kê các cách xếp ba bạn An, Bình, Cường vào ba vị trí khác nhau. Giả sử các vị trí là 1 , 2 , 3 1, 2, 3 1 , 2 , 3
An, Bình, Cường An, Cường, Bình Bình, An, Cường Bình, Cường, An Cường, An, Bình Cường, Bình, An Tổng cộng có 3 ! = 6 3! = 6 3 ! = 6
7. Các lỗi thường gặp khi liệt kê và cách tránh Bỏ sót trường hợp: Do liệt kê không theo hệ thống. Khắc phục: Liệt kê theo thứ tự, phân loại rõ ràng. Lặp lại trường hợp: Ghi hoặc đếm trùng phần tử. Khắc phục: Đánh dấu các trường hợp đã liệt kê, kiểm tra lại các phần tử. Nhầm lẫn điều kiện: Chưa đọc kỹ yêu cầu đề bài dẫn đến liệt kê sai. Khắc phục: Xác định rõ ràng điều kiện trước khi bắt đầu. 8. Tóm tắt và các điểm chính cần nhớ Liệt kê là kỹ năng quan trọng giúp học tốt các phần đếm, xác suất, tổ hợp. Cần xác định rõ quy tắc, điều kiện liệt kê và thực hiện theo hệ thống để không bỏ sót hay lặp lại. Liên hệ chặt chẽ với các khái niệm: tập hợp, xác suất, tổ hợp, hoán vị, chỉnh hợp. Nên luyện tập nhiều dạng bài để thành thạo kỹ năng này. Liệt kê là bước khởi đầu cơ bản để tiến sâu hơn vào các chủ đề lớn của toán học lớp 10 và nhiều lĩnh vực liên quan khác.
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