求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1);(2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)

Question

求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1); (2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)
求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)
(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)
排列组合

Answer 1

（1）A(n+1,n+1) = (n+1)!= (n+1)*n*...*2*1
所以题目左边 = (n+1)!-(n)!= (n+1-1)*(n)!= (n*n)*(n-1)!= 右边,得证
（2）把右边的每个数都写成C(m,n) = n!/(m!*(n-m)!)的形式,
右边（字母太多看着也烦,就不列了）通分成分母为(m!*(n-m+1)!)的形式
右边 = ( m*n!+ (n-m)*(n-m+1)*(n-1)!+ (n-m+1)*m*(n-1)!))/(m!*(n-m+1)!)
= ( (n+1)!)/(m!*(n-m+1)!)
= 左边
命题得证.