$$\sum _{k=0}^n\left(PP^k_n\cdot C_n^k\right)$$
So the sum is co-prime cases and the multiply is something that tells me that for each choice of k in n items (with order without repeats) there is choice of 5 elements with length of chosen k.
I tried something like:
Choosing committee of k in n people, which play lottery with the enumerated balls 1,2,3,4,5 of length of chosen committee.
My final mission is to simplify such an expression for something else and I tried ti find a problem, which I can describe in words and try to find another combinatorical expression, which is more simple then, what was I given:
$$\sum _{k=0}^n\:\left(5^k\right)\left(_k^n\right)$$