From 6 vertices we select 4 vertices in 6C4 = 15 ways.
Now, with these 4 vertices, we can form only 3 distinct cycles.
How ? Since, all 4 vertices have adjacent edge to other 3 choosen vertices, i.e. total 6 edges. Now every cycle is 2-regular. Therefore, every vertice is adjacent to 2 other vertices. So, for first vertex, we have 3 choices to choose 2 adjacent vertices out of 3 vertices. Now, we have drawn 2 lines of cycle and selected 3 vertices out of 4. For last, we have no choice, since first vertex already rejected it to be its adjacent. so, it will join to other vertices.
Hence 15*3=45 is the answer.