For N greaterthanorequalto 3 find the number of distinct per

For N greaterthanorequalto 3, find the number of distinct perfect matchings in the cycle C_n of length n. Justify your answer.

Solution

If G has a perfect matching, then G has an even number of vertices, so C(2k1) has
no perfect matchings.
Any perfect matching of C2k must either use the edge (1, 2) or the edge
(1, n).
In the first case the unique perfect matching uses (2i 1, 2i), for i = 1, . . . , k, and
in the latter case the unique perfect matching uses (1, n) and (2i, 2i + 1), for i = 1, . . . , k 1.
Therefore the number of perfect matchings of Cn is 0 if n is odd and 2 if n is even.
and therefore number of distinct perfect matching is 2.