Introduction to Probability · Problem 35

Combinatorics — Grinstead & Snell

Mathematics — Probability · binomial-identities

Problem

Prove the following binomial identity

\binom{2n}{n} = \sum_{j=0}^{n} \binom{n}{j}^2.

Hint: Consider an urn with

n

red balls and

n

blue balls inside. Show that each side of the equation equals the number of ways to choose

n

balls from the urn.