Exercise 1547Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

Pentagon identity

#pentagon#Pascal#binomial

Statement

Show that $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ and deduce a "pentagon" identity around a small pentagon in Pascal's triangle.

Hint

The pentagon is formed by five adjacent entries; the identity relates the two "legs" to the "top".

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Pascal. The basic identity $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ .

Iterate. Applying it at each vertex of a pentagon gives a relationship among five adjacent entries.

Conclusion. Various pentagon identities follow from iterating Pascal's identity.