[JEE Advanced 2000] prove binom nm binom n 1 m binom mm binom n 1 m 1

Prove $\binom nm+\binom{n-1}m+\cdots+\binom mm=\binom{n+1}{m+1}$, and hence $\binom nm+2\binom{n-1}m+\cdots+(n-m+1)\binom mm=\binom{n+2}{m+2}$.