Exercise 1546Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

Hexagon identity

#hexagon#Pascal#binomial

Statement

Show that $\binom{n-1}{k-1}\binom{n}{k+1}\binom{n+1}{k}=\binom{n-1}{k}\binom{n}{k-1}\binom{n+1}{k+1}$ .

Hint

Expand using the factorial formula.

Show solutionHide solution

LHS. $\frac{(n-1)!}{(k-1)!(n-k)!}\cdot\frac{n!}{(k+1)!(n-k-1)!}\cdot\frac{(n+1)!}{k!(n+1-k)!}$ .

RHS. $\frac{(n-1)!}{k!(n-1-k)!}\cdot\frac{n!}{(k-1)!(n-k+1)!}\cdot\frac{(n+1)!}{(k+1)!(n-k)!}$ .

Compare. Both contain $(n-1)!n!(n+1)!$ in the numerator and various factorials in the denominator. Direct expansion shows both equal.

Conclusion. The hexagon identity holds. (The identity is so named because it traces a hexagon on Pascal's triangle.)