An identity with squares

#squares#binomial#double counting

Statement

Show that $\sum_{k=0}^{n}\binom{n}{k}^2=\binom{2n}{n}$ .

Hint

Use Vandermonde with $m=p=n$ and $r=n$ .

Show solutionHide solution

See "Sum of squares of binomial coefficients" in this category.

Conclusion. $\sum_k\binom{n}{k}^2=\binom{2n}{n}$ .