Vandermonde’s identity is a famous relationship between binomial coefficients. It has the following form:
Blitzstein and Hwang, in their book Introduction to Probability, give a story proof to the identity. Suppose m boys and n girls want to form a committee of k members; there are (m+n)Ck possibilities for doing it. It forms the left-hand side of the identity.
For every j number of boys in the committee, there will be (k-j) girls in it. The number of possibilities is mCj x nCk-j. The required possibility will be the sum of all those possibilities where j ranges from 0 (no boys) to k (only boys).
