So I have part of this solved ( I think)

The basics:

Total population of objects :20

slots to fill: 5

total population of sets: 3.2MM (20^[SUP]5[/SUP])

My goal is to organize this total population into groups.

aaaaa - 5 of a kind

aaaab - 4 of a kind, 1 single

aaabb - 3 of a kind, 1 pair

aaabc - 3 of a kind, 2 singles

aabbc - 2 pairs, 1 single

aabcd - 1 pair, 3 singles

abcde - all singles

so using [SUB]n[/SUB]P[SUB]r[/SUB] I came up with the following:

category-count

aaaaa-20

aaaab-380

aaabc -6,840

aaabb-380

aabbc -6,840

aabcd -116,280

abcde -1,860,480

This is incomplete as it sums to 1,991,220.

the next step as I was seeing it was that there are variations of each category:

Example: aaaab,aaaba,aabaa and so on.

the plan was to calculate the permutations for each and then multiplying that by the initial calculations. but this yields much too large a population. so now I am stuck.

Here is the google sheets document I have been working with:

https://docs.google.com/spreadsheets...it?usp=sharing

Any hints and help are greatly appreciated!