Combinatorial Identities

Lesson · Intermediate

Combinatorics

\binom{k}{k} + \binom{k+1}{k} + \binom{k+2}{k} +\cdots+ \binom{n}{k} = \binom{n+1}{k+1}

\binom{m}{0}\binom{n}{k} + \binom{m}{1}\binom{n}{k-1} + \binom{m}{2}\binom{n}{k-2} +\cdots+ \binom{m}{k}\binom{n}{0} = \binom{m+n}{k}

\binom{k}{k} + \binom{k+1}{k} + \binom{k+2}{k} +\cdots+ \binom{n}{k} = \binom{n+1}{k+1}

\binom{m}{0}\binom{n}{k} + \binom{m}{1}\binom{n}{k-1} + \binom{m}{2}\binom{n}{k-2} +\cdots+ \binom{m}{k}\binom{n}{0} = \binom{m+n}{k}