Combination problem, what's wrong with my reasoning?

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The discussion centers on a combinatorial problem involving forming a team of 4 people from 4 women and 7 men, requiring at least 2 women. The initial approach incorrectly calculates the number of combinations by choosing 2 women and then 2 from the remaining individuals, resulting in an erroneous total of 216 ways. The mistake arises from overcounting teams that contain the same members in different orders. Specifically, teams with 3 women are counted multiple times due to different selection sequences. The correct reasoning must account for these repetitions to arrive at the accurate number of combinations.
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Form a team by choosing 4 persons among 4 women and 7 men. You need at least 2 women in the team. How many ways to do it?

My solution:
1. Choose 2 women among the 4 availaible women. There is 2C4 ways of doing it.
and
2. Choose 2 persons among the 9 remaining people, there is 2C9 of doing it.

2C4 * 2C9 = 216 ways.

This is wrong, but I can't see why...
 
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hi fishingspree2! :smile:
fishingspree2 said:
This is wrong, but I can't see why...

because you're counting some teams more than once

eg if the team contains exactly 3 women, A B and C, then you're counting that team 3 times:

once for A and B being chosen first, once for B and C, and once for C and A :wink:
 

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