(a) There is an even number 2 n of people at a party, and...

(a) There is an even number $2 n$ of people at a party, and they talk together in pairs, with everyone talking with someone (so $n$ pairs). In how many different ways can the $2 n$ people be talking like this? (b) Now suppose that there is an odd number $2 n+1$ of people at the party with everyone but one person talking with someone. How many different pairings are there?

(a) There is an even number $2 n$ of people at a party, and they talk together in pairs, with everyone talking with someone (so $n$ pairs). In how many different ways can the $2 n$ people be talking like this?
(b) Now suppose that there is an odd number $2 n+1$ of people at the party with everyone but one person talking with someone. How many different pairings are there?

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Key Concepts

Perfect Matching

A perfect matching is a set of pairs in which every element of a set is paired with exactly one other element, with no element left unpaired. In combinatorics, counting the number of perfect matchings in a set with an even number of elements (such as pairing 2n people) is a classic problem, as it requires forming disjoint “pairs” that cover the entire set without overlap.

Double Factorial

The double factorial is a product obtained by multiplying a sequence of integers separated by two. Specifically, for an odd integer (2n-1)!!, it means multiplying all odd numbers from 1 up to (2n-1). This concept arises naturally in counting the number of perfect matchings on a set of 2n elements, since the recursive method of pairing leads to the product (2n - 1) * (2n - 3) * ... * 1.

Combinatorial Selection

Combinatorial selection involves choosing a subset from a larger set according to some criteria. When an odd number of elements is involved, one common approach is to first select the element that remains unpaired and then form perfect matchings on the remaining even number of elements. This two-step process – selecting the unpaired element and then counting the perfect matchings among the rest – is a fundamental technique in problems where complete pairing is not possible.

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