✔ 最佳答案
m! / [n!(m - n)!] = mCn , m>n.
We are required to prove mCn is an integer.For m = 2 , then n = 1 , 2C1 = 2 is true.
Assuming when m = k , kCn are integers for k > n = 1,2,3,...,k-1When m = k+1 , m > r = n+1 = 2,3,4,...,k
(k+1)C r
= (k + 1)! / [r! (k - r + 1)!]
= k! (k - r + 1 + r) / [r! (k - r + 1)!]
= k! / [r!(k-r)!] + k! / [(r-1)!(k-r+1)!]
= kCr + kC(r-1)
= kC(n+1) + kCnBy assumption, kCn are integers, also kC(n+1) are integers since kCk = 1.Thus, if kCn are integers, then (k+1)C r are integers. Proved.