Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME Find the answer

Given a set $A$ with $n \geq 1$ elements, find the number of consistent 2-configurations of $A$ of order 1 with exactly 1 cell.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

There must be some pair $\{a, b\}$ in the 2-configuration, since each element $a \in A$ must belong to one pair. Since neither $a$ nor $b$ can now belong to any other pair, this must be the entire cell. Thus, there is 1 such 2-configuration when $n=2$ , and there are none when $n \neq 2$ .

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