排列组合公式求证：Anm+mAnm-1=An+1m

Question

排列组合公式 求证：Anm+mAnm-1=An+1m

Answer 1

Anm+mAnm-1
= n!/(n-m)! + m*n!/(n-(m-1))!
= (n-m+1)n!/(n-m+1)! + m*n!/(n-m+1)!
= {(n-m+1) + m}n!/(n-m+1)!
= (n+1)*n!/(n-m+1)!
= (n+1)!/[(n+1)-m]
= An+1m