Board Thread:General Discussion/@comment-30334719-20170222043910/@comment-26335449-20170602145422

Sorry for the late reply! This just popped up when googling some stuff and I got clickbaited.

Your question is rather ambiguous and there are 2 different interpretations of the problem: an easy one and a hard one. I'll answer both.

The EASY one: The number of each girl doesn't matter (i.e; 2x honk and 7x eli is the "same" combination as 5x honk and 4x eli).

This one is simple. The number of ways to make an n-girl team is ncr(9,n), for an integer n in [1,9].

So...

ncr(9,1) + ncr(9,2) + ... ncr(9,9) = 511

So if this is the problem you're asking, the answer is 511.

The HARD one: The number of each girl DOES matter (i.e; 2x honk and 7x eli is DIFFRENT from 5x honk and 4x eli.)

This is a basic combinatorics problem. We need to apply the multichoose theorem, also called the stars and bars' theorem. Here's how it works:

Order the 9 girls. I'll order them like this: Rin - Hanayo - Maki - Honk - Umi - Kotori - Eli - Nico - Nozomi

Any team can be represented by |'s and x's, where each x is a girl and a | is a partition between types of girls:


 * |XX| | | |XXX|XXX|X = 2x maki, 3x eli, 3x nico, 1x nozomi

XXXXXXXX| | | | | | |X| = 8x rin, 1x nico


 * | | | |XXXXXXXXX| | | = 9x kotori

So the problem becomes: How many ways can we arrange 8 |'s and 9 x's into a string of length 17? The answer is obviously ncr(17,8) = ncr(17,9) = 24,310. Good luck counting that!

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Thanks, hope this helped!