Competition problem — Combinatorics

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Combinatorics Difficulty 3.8 AMC 10/12 Find the answer United States

Problem:
Find the number of triangulations of a general convex $7$ -gon into $5$ triangles by $4$ diagonals that do not intersect in their interiors.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Define the Catalan numbers by $C(n) = \frac{1}{n+1} \binom{2n}{n}$ . The current solution is the $C($ number of triangles $) = C(5) = 42$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.