There can be total ^{6}C_{4} ways to pick 4 vertices from 6. The value of ^{6}C_{4} is 15.
Note that the given graph is complete so any 4 vertices can form a cycle.
There can be 6 different cycle with 4 vertices. For example, consider 4 vertices as a, b, c and d. The three distinct cycles are
cycles should be like this
(a, b, c, d,a)
(a, b, d, c,a)
(a, c, b, d,a)
(a, c, d, b,a)
(a, d, b, c,a)
(a, d, c, b,a)
and
(a, b, c, d,a) and (a, d, c, b,a)
(a, b, d, c,a) and (a, c, d, b,a)
(a, c, b, d,a) and (a, d, b, c,a)
are same cycles.
So total number of distinct cycles is (15*3) = 45.
The answer is 45.