What in the Binomial Theorem?

What this binomial-coefficient identity proof tests

This is a medium-difficulty combinatorial proof question that asks you to establish a fundamental identity relating sums of squared binomial coefficients to a single central binomial coefficient. Questions of this kind appear regularly in quant interview loops because they reward both algebraic fluency and combinatorial intuition.

The problem invites multiple solution approaches: a direct algebraic expansion using the binomial theorem, a combinatorial counting argument that interprets both sides as ways to select from a population, or a convolution perspective. Interviewers are typically interested in which method you choose, how clearly you justify each step, and whether you can explain why the identity holds conceptually, not just verify it algebraically.

• Binomial theorem and polynomial identities
• Combinatorial interpretations of binomial coefficients
• Convolution and symmetry arguments
• Vandermonde's identity and related summation identities