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Imagine a parliament with 4 parties that can form a government (this is a parliamentary democracy, not the US style democracy). The government does not need to have a majority behind it in the parliament (in the long run it does need it but not for this example).

How many possible combinations of parties are there?

I thought you calculated this using permutations but 4/4 returns 24 and when I do this manually I only come up with 15 combinations:

ABCD ABC ABD ACD BCD AB AC AD BC BD CD A B C D

https://www.calculatorsoup.com/calculators/discretemathematics/permutations.php?n=4&r=4&action=solve

In my case order does not matter (that is, ABC and CBA is the same) and items must not repeat. Does a regular permutation calculation take this into account? If not, how do I calculate this? Something like "combination (4 elements out of of 4 elements) + combination (3 of 4) + combination (2 of 4) + combination (1 of 4)"? What is this called?

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You can consider this as the power set of $\{A, B, C, D\}$ minus the empty set. This gives $2^4 - 1 = 15$ sets as required.

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  • $\begingroup$ What is the relationship between "power sets" and permutations/combinations? I remember I had this function on my calculator (HP) during high school. $\endgroup$ – d-b Sep 12 '18 at 12:32

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