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1985年全国高中联赛有一道求不定方程整数解的竞赛题,原题如下: 方程2x_1+x_2+x_3+…+x_(10)=3共有多少组不同的非负整数解? 此题难度不大,但其一般化以后的结论却是很有意思的,下面先证明两个关于不定方程整数解的命题。命题1 不定方程 x_1+x_2+…+x_m=n (n≥m)共有C_(n-1)~(m-1)=1组不同的正整数解。 (证明请参看苏淳编写的“同中学生谈排列组合”一书。) 命题2 不定方程 x_1+x_2+…+x_m=n(n≥0)共有C_(n+m-1)~(m-1)组不同的非负整数解。
In 1985, the national high school league had a contest for the integer solution of the indefinite equation. The original question is as follows: Equation 2x_1+x_2+x_3+...+x_(10)=3 How many sets of different non-negative integer solutions are there? This question is not difficult, However, its generalized conclusions are very interesting. We first prove two propositions about the integer solutions of indefinite equations. Proposition 1 Indefinite equation x_1+x_2+...+x_m=n (n≥m) There are altogether C_(n-1)~(m-1)=1 sets of different positive integer solutions. (For the proof, please refer to the book “Same Middle School Student Talk Arrangement” written by Su Shi.) Proposition 2 Indefinite Equation x_1+x_2+...+x_m=n (n≥0) There are C_(n+m-1)~(m-1) ) Different non-negative integer solutions for a group.