Mathematical Induction

2014-03-21 6:12 pm

Prove by M. I. that m! is divisible by n!(m - n)! where both m and n are positive integers and m > n.

回答 (2)

2014-03-22 1:07 am
✔ 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.
2014-03-22 1:51 am
Dear Anonym,

here is another reference:

http://11235813.wikidot.com/algebra:20110523-ncr-is-an-integer


收錄日期: 2021-04-21 22:30:12
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20140321000051KK00019

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