number of ways to label in a cycle

Question

Suppose I have a 6 vertices complete graph, Say it G. It is labeled, Now I need to find all distinct 4 vertices cycles. So for first step there are 6C4 ways to select 4 length cycles , but as it is labeled I need to consider labeling also , stuck here. What are no. of ways in which 4 length cycle can be labelled ?

Answer 1

Hint: Selecting the vertices (6C4) doesn't specify the cycle. You could have $1 \to 2 \to 3 \to 4$ or $1 \to 3 \to 2 \to 4$ The easy way is just to pick the four vertices in order, because any series of four makes a cycle-how many ways can you do that? Then recognize that the same cycle can be picked in a number (what number?) of ways.