證明n+1必可整除C(2n,n)?

考慮 C(2n, n + 1)
= (2n)! / [(n + 1)! (n - 1)!]
= n (2n)! / [(n + 1) n! n!]
= n/(n + 1) (2n!)/(n! n!)
= n/(n + 1) C(2n, n)
= [1 - 1/(n + 1)] C(2n, n)
= C(2n, n) - C(2n, n)/(n + 1)

因此
C(2n, n)/(n + 1)
= C(2n, n) - C(2n, n + 1)
= C(2n, n) - C(2n, n - 1) 是一個正整數

即 C(2n, n) 可被 (n + 1) 整除。

【請看以下有用的參考資料。】