Total number of possible ways to repair N damaged components with M number of crews

Question

Assuming there are "N" damaged components in a electric power network after an earthquake, and there are "M" number of crews out there that can work on the damaged components, and assuming that each crew can work on "P" number of damaged components simultaneously, what is the total number of possible resource allocation/restorations?

I believe there are N! ways of going through the damaged components like one-by-one ("M" = "P" = 1), and I suppose (I am not sure) that the total number of ways for allocating/distributing "M" crew between the "N" damaged components can be calculated as N!/M!(N-M)! (if this is true, I would aprreciate if you could explain why as I have zero experience in this area), but not only I am not sure about this but also I do not know how to include "P" in the solution!

Answer 1

Absent better definition of the question, I will assume

1. $N=kMP$, so there are $k$ rounds of assigning repairs to each crew. Each crew gets a full list of assignments each time
2. The crews are interchangeable, so assigning $1,2,3$ to the first crew and $4,5,6$ to the second is the same as assigning $4,5,6$ to the first and $1,2,3$ to the second
3. It matters which round a particular item is repaired

We can line up the $N$ items to repair in $N!$ ways, then break them into blocks of $P$ for assignment to crews. The first $M$ blocks are repaired in the first round, the next $M$ in the second, and so on up to round $k$. We ask how many different orders of the items result in the same assignment. Each block can be reordered in $P!$ different ways so we divide by $(P!)^{\frac NP}$ The $M$ blocks in a round can be rearranged among the crews in $M!$ ways so we divide by $(M!)^k$ This gives $$\frac {N!}{(P!)^{\frac NP}(M!)^k}$$ different assignments.