How many perfect matchings are there in a complete graph of 6 vertices ?

Answer: 15

Formula: The number of perfect matchings in a complete graph on 2n vertices is (2n)! / (2^n · n!). For 6 vertices (n = 3):

• (6)! / (2^3 · 3!) = 720 / (8 · 6) = 720 / 48 = 15

• Alternative reasoning: Pair a fixed vertex with any of 5 choices, then pair one of the remaining vertices with any of 3 choices, then 1 choice remains. This gives 5 · 3 · 1 = 15.

• Thus there are 15 perfect matchings.