Pairing up six people

Question

Six people are to be divided into three pairs (for example, to play three games of chess). The pairs are not labelled: it does not matter which pair is "first", only who is with whom. How many different ways are there to pair them up? Then find the number for 8 people and the general formula.

Accepted Answer

15 ways for six people; 105 for eight. The recursive count. Take one particular person, say the oldest. Their partner can be any of the other 5 people. Once that pair is fixed, four people remain; take the oldest of them, whose partner can be any of the other 3; then two people remain and form the last pair in 1 way. Total: 5 × 3 × 1 = 15. Eight people. 7 × 5 × 3 × 1 = 105. Ten people: 9 × 7 × 5 × 3 × 1 = 945. The general formula for 2n people is (2n minus 1) × (2n minus 3) × ... × 3 × 1, written (2n minus 1)!! and called a double factorial. It also equals (2n)! divided by (2 to the n × n!): …

Pairing up six people

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