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How many ways can a group of 12 including 4 boys and 8 girls be formed into two 6 person volleyball teams with no restrictions?
462 ways
Explanation
In the question, it does not seem to matter how many boys or girls are on a team. So really you are asking how many ways are there to divide 12 people into two teams of 6.
The answer is (12C6)/2 where 12C6 means 12 choose 6
First 12 chose 6 is 12!/(12-6)!(6)! which is 12x11x10x9x8x7x6x5x4x3x2x1/(6x5x4x3x2x1)(6x5x4x3x2x1)=
(12x11x10x9x8x7)/(6x5x4x3x2x1)=924
(As a side note, the Google browser will calculate these numbers if you just type in 12 choose 6, for example)
Now once one team of 6 is chosen, the other team is automatically the remaining 6 so we have really double counted. That is why we divide by 2 and the answer
is 462 ways.
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