Why doesn't this work (combinations and permutations)?
There is a group of 13 people and you are to choose 10 of them (order doesn’t matter) – trivially, this is 13C10 = 286.
Now say 3 of the people are women and you are to choose a team that includes at least one woman. This is equal to the total number of groups minus the one grouping that is only men, 285.
Another way I thought about the problem doesn’t yield the right answer even though it seems to make sense. It goes as follows: choose one woman to include on the team – 3C1 – then choose any grouping of the remaining people – 12C9. Multiply and get 660, which is obviously wrong because it’s larger than the number of possible teams even when you place no restrictions on your choice-making parameters.
Why is this method wrong?
This question is in the General Section. Responses must be helpful and on-topic.