(Generalized Vandermonde's Identity) Show that $$\sum_{k_1+\cdots+k_p=m}\binom{n_1}{k_1}\binom{n_2}{k_2}\cdots\binom{n_p}{k_p}=\binom{n_1 + \cdots + n_p}{m}$$
Show that $$\sum_{k=0}^m \binom{n}{k}\binom{n-k}{m-k}= 2^m\binom{n}{m}$$
(Generalized Vandermonde's Identity) Show that $$\sum_{k_1+\cdots+k_p=m}\binom{n_1}{k_1}\binom{n_2}{k_2}\cdots\binom{n_p}{k_p}=\binom{n_1 + \cdots + n_p}{m}$$