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有一個問題:“以20冊數學通報任意分配給37個圖書館,有多少種方法?”這個問題的解決,一般說來,與下面所述是完全相同的,即:設有p個正整數r_1,r_2,r_3,…r_p,其中可以有零和相等的,不過,它們之間有一個關係式r_1+r_2+…+r_p=n…(1) 存在,n是一個給定的正整數,則能適合(1)的r_1,r_2,…,r_p的組數為H_n~p=C_(n+p-1)~p。 現在把這結果稍加推廣:設有p個正整數r_1,r_2,…,r_p,其中可以有相等的,但是每一個都不准小於一個給定的正整數a,而且它們之間仍有關係式(1)存在,n是一個給定的不小於p·a的正整數,試求能適合(1)的r_1,r_2,…,r_p的組數。 關於這個問題,我們這樣來討論:依假設,r_1,r_2,…,r_p都不准小於a,也就是說,它們的值至少是a。
There is a question: “With 20 books of mathematics assigned to 37 libraries randomly, how many methods are there?” The solution to this problem is, in general, the same as described below, ie, there are p positive integers. R_1,r_2,r_3,...r_p, where there may be zeros and equals, but there is a relation between them r_1+r_2+...+r_p=n...(1) exists, n is a given positive integer, then The number of sets of r_1, r_2, ..., r_p suitable for (1) is H_n~p=C_(n+p-1)~p. Now this result is slightly generalized: There are p positive integers r_1, r_2, ..., r_p, which can be equal, but each is not allowed to be less than a given positive integer a, and there is still a relationship between them. Equation (1) exists. n is a given positive integer not less than p·a. Try to find the number of r_1, r_2,..., r_p that can fit in (1). On this issue, we discuss this way: According to the hypothesis, r_1, r_2, ..., r_p are not allowed to be smaller than a, that is, their values are at least a.