※ 引述《pcpo8992 (E~T shot)》之銘言：
: m,n 屬於 Z
: show that (2m)!(2n)!/m!n!(m+n)! is integer
: 有用數學歸納法去做但ㄧ直卡住 想請板上神人幫忙
: 謝謝
質數p在(2m)!(2n)!/m!n!(m+n)!的標準分解式的次數為
sum ( [2m/q]+[2n/q]-[m/q]-[n/q]-[(m+n)/q] )
q=p^i
WLOG, assume 0<=m<=n<q.
then [2m/q]+[2n/q]-[m/q]-[n/q]-[(m+n)/q]
= [2m/q]+[2n/q]-[(m+n)/q] >=0 because 2n>=m+n.
Hence the sum >=0. So, (2m)!(2n)!/m!n!(m+n)! is an integer.
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