How many perfect matchings are there in a complete graph of 10 vertices?(a) 60(b) 945(c) 756(d) 127I got this question in homework.This intriguing question originated from Isomorphism in Graphs in division Graphs of Discrete Mathematics

How many perfect matchings are there in a complete graph of 10 vertice

1.	How many perfect matchings are there in a complete graph of 10 vertices?(a) 60(b) 945(c) 756(d) 127I got this question in homework.This intriguing question originated from Isomorphism in Graphs in division Graphs of Discrete Mathematics
Answer» The correct OPTION is (b) 945 For EXPLANATION: Perfect matching is a set of edges such that each vertex APPEARS only once and all vertices appear at least once (exactly ONE appearance). So for n vertices perfect matching will have n/2 edges and there won’t be any perfect matching if n is odd. For n=10, we can choose the first edge in ^10C2 = 45 ways, second in ^8C2=28 ways, third in ^6C2=15 ways and so on. So, the total number of ways 45281561=113400. But perfect matching being a set, order of elements is not important and the permutations 5! of the 5 edges are same only. So, total number of perfect matching is 113400/5! = 945.

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