经典排列公式的正向推导

经典排列公式的正向推导

C(n+k+1,k+1)-C(n+k,k)=C(n+k,k+1)

将组合式展开成计算式：

C(n+k+1,k+1)={(n+k+1)*(n+k)...*(n+1)}/(k+1)!

同样地:

C(n+k,k)={(n+k)*(n+k)...*(n+1)}/(k)!

GO

C(n+k+1,k+1)-C(n+k,k)={(n+k+1)*(n+k)...*(n+1)}/(k+1)!   -  {(n+k)*(n+k-1)...*(n+1)}/(k)!

Go

C(n+k+1,k+1)-C(n+k,k)={(n+k+1)*(n+k)...*(n+1)}/(k+1)!   -  {(n+k)*(n+k-1)...*(n+1)*(k+1)}/(k+1)!

GO

合并：

C(n+k+1,k+1)-C(n+k,k)={(n+k+1)*(n+k)...*(n+1)-(n+k)*(n+k-1)...*(n+1)*(k+1)}/(k+1)!  

GO

C(n+k+1,k+1)-C(n+k,k)={{(n+k+1)-(k+1)} *(n+k)*(n+k-1)...*(n+1) }/(k+1)!  

Go

C(n+k+1,k+1)-C(n+k,k)={n *(n+k)*(n+k-1)...*(n+1) }/(k+1)!  

GO

改变形式n移动位置有：

C(n+k+1,k+1)-C(n+k,k)={(n+k)*(n+k-1)...*(n+1)*n }/(k+1)!  

Go

C(n+k+1,k+1)-C(n+k,k)=C( n+k , k+1).