# Help checking a question on graph theory.

Can someone check these (bit skeptical of my answers).

a) How many copies of $C_4$ in $K_n$?

Picking any 4 vertices can be used to give a copy of $C_4$of each of these there are $4!$ ways in which the vertices can be arranged but this overcounts by a factor of 8 (4 vertices $\times$ 2 directions that can be transversed) so the total is $(4!/8)* nC4=3(nC_4)$.

b) How many copies of $P_3$ in $K_n$?

Any 4 vertices choose represent $4C3 = 4$paths so we have 4(nC4) in total.

How can the answer to a be smaller than the answer to b when we have that every$P_3$ in $K_n$ is contained within a $C_4$ in $K_n$?

For the second question there are $\binom{n}{3}$ ways to select the three vertices in the path. And after that there are $3$ ways to select which of the $3$ vertices is going to be the middle one, so $3\binom{n}{3}$