數論基本題

(M+1)(M+2)(M+3)...(M+N) / N!
= (M+N)! / (M! N!)
= C(M+N,N)
換句話說是要證明C(n,r)為正整數
用數學歸納法,定義P(n)為: C(n,r)都是正整數,其中 0 <= r <= n
若n = 1, C(1,0) = C(1,1) = 1很明顯都是正整數
假設 P(k)為真,即 C(k,0), C(k,1), C(k,2) ... ,C(k,k)都是正整數
則C(k+1,p) = (k+1)! / [(k+1-p)! p!] 其中1 <= p <= k
= {k! / [(k-p)!(p-1)!]} [(k+1) / (k+1-p)p]
= {k! / [(k-p)!(p-1)!]} [1/(k+1-p) + 1/p]
= k! / [(k+1-p)!(p-1)!] + k! / [(k-p)!p!]
= C(k,p-1) + C(k,p)為正整數
另C(k+1,0) = 1; C(k+1,k+1) = 1都是正整數
=> P(k+1)為真
換句話說C(n,r)皆為正整數

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