Show that (2k+1)!(2n)!/(n!k!(n+k+1)!) is always an integer.
Assuming k and n are non-negative integers, then there are three possibilities, 'big n', 'big k' and n=k
Starting with 'big n', assume that n>=(k+1), then n!, k!, and (n+k+1)! all divide (2n!) severally.
If n=k, then (2k+1)! is bigger than 2n! then n!, k!, and (n+k+1)! all severally divide that instead.
For 'big k', i.e. k>n, then n!, k!, and (n+k+1)! all divide (2k+1)! severally.
Thus the given expression is always an integer.
Note1: This is not necessarily true if k and n are themselves non integers, or negative numbers.
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Posted by broll
on 2023-09-11 22:20:40 |
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