Let be non-negative integers with .

MATHEMATICS · Multi-part · CBSE · Mathematics · Permutations and Combinations · Permutations and Combinations

Problem

Let

m,n,r

be non-negative integers with

0\le r\le m+n

. (i) Evaluate numerically, for

m=4,\;n=3,\;r=3

, the sum

\sum_{k=0}^{3}{4\choose k}{3\choose 3-k}

and compare it with

{7\choose 3}

. (ii) Prove, for general

m,n,r

, the identity

\sum_{k=0}^{r}{m\choose k}{n\choose r-k}={m+n\choose r}.