Among the various subgraphs of $K_5$, how many are cycles? Among the various subgraphs of $K_5$, how many are cycles?
I know the answer is $37$ because the number of $3$-cycles is $10$, the number of $4$-cycles is $15$, and $5$-cycles is $12$. Could anyone explain in detail how these numbers are reached?

First notice that any $n \in \{3,4,5\}$ vertices you choose from $K_5$, you can make a cycle out of them. When you are counting the $n$-cycles, you need to count the number of ways you can choose $n$ vertices out the given $5$ vertices, and for each of those choices you need to count how many ways those $n$ vertices can be arranged into a cycle. 
So when counting the $3$-cycles, since there is only one way to have a cycles with $3$ vertices it would be 
$$
  \binom{5}{3} \times (1) = 10 \times (1) = 10
$$ 
The cases for $4$ and $5$ vertices is done similarly. You've just got to figure out how to count the number of ways $4$ and $5$ vertices can be connected into a cycles. Can you take it from here?
