Question

求证C(m,n-1)+C(m,n-2)+C(m,n-3)+...+C(m,m+1)+C(m,m) =C(m+1,n)

Answer 1

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

这个式子可以直接验证,也可以算两次得证.

然后递推

C(m+1,n)

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

=C(m,n-1)+C(m,n-2)+C(m+1,n-2)

...

=C(m,n-1)+C(m,n-2)+C(m+1,n-2)+...+C(m,m+1)+C(m+1,m+1)

=C(m,n-1)+C(m,n-2)+C(m+1,n-2)+...+C(m,m+1)+C(m,m)