Let $K_n$ be the complete graph of order $n$ and $P_m$ a path with $m$ distinct vertices, $1 \leq m \leq n$.
Question: How many distinct copies of $P_m$ are contained in $K_n$?
Given that a permutation maps a path to a different path it seems like there will always be another permutation which will send the original path to the same path, different from the original, so that the number of copies of $P_m$ contained in $K_n$ will be:
Is this correct? If not, or if so, could someone provide a more rigorous derivation of the correct value?