I have $4$ electrons to place in $7$ orbitals. Each orbital can hold up to some maximum number of electrons. Let's name the orbitals $a,b,c,d,e,f,g$ for reference. Let's say the maximums are $1,1,2,5,2,1,1$.
So for instance, I could place one electron in orbitals $a,b,f,g$ or I could place all four in orbital $d.$ Or, I could place two in both $c$ and $e$, but I cannot place more than one in $a,b,f,g,$ nor more than two in $c$ or $e$, etc.
How many combinations are there?
The basis for this problem - I've written a program to generate electron configurations which are bounded by maximum occupancy. I want to check that my program is generating all possible configurations, by comparing the actual length of the list with this calculation. My program says there are $90$ (edit: not $175$!) configurations, which may or may not be true.