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I'm trying to figure out how many different combinations exist in some sets, after doing research I've been thinking that I've only been calculating the number of permutations but I'm starting to think maybe it is actually the number of combinations.

There are 8 different sets.

1: 57 items
2: 5 items
3: 6 items
4: 2 items
5: 9 items
6: 4 items
7: 7 items
8: 2 items

I am picking one item from each set, and trying to figure out how many different combinations there are.

Straight multiplying those numbers, gives me 1,723,680. From what I previously thought, that would be the number of permutations.

However, If we were trying to find combinations of { A, B, C } {D, E, F} I see that multiplying appears to give the number of combinations: 3*3=9 {AD, AE, AF, BD, BE, BF, CD, CE, CF}

I've been looking at the n! / k!(n-k)! formula, but have a lot trouble applying it to my situation with several sets and different number of items in each set.

Is 1,723,680 the number of permutations or combinations? And, if you think you are able to tell me, where is my confusion stemming from?

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The product of all of the set sizes is the number of combinations. Order doesn't matter (you pick one from each set and throw them in a pile).

To find the number of permutations in this case, you multiply the number of combinations by $8!$. (I'm assuming all $92$ objects are distinct.) After you've picked your eight, you have $8!$ ways to line them up.

The combination formula you wrote gives the number of ways you can pick $k$ distinct items out of $n$, without considering order. This doesn't apply here.

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