Counting Cycles - Intuition or formula

Question

How many cycles of length 4 are there in K5? My intuition is 5 (using 5 choose 4). However, I know the answer is 15. I'm trying to understand the intuition behind it.

Thanks!

Answer 1

Your intuition is correct as you can choose $4$ vertices in $\binom{5}{4}$ ways.
Further, for each there are $\dfrac{1}{2}(4-1)!$ free circular permutations.