the ways of chossing 4 people from 12 is the same as ways of choosing 8 fromĀ 12. why?

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Matthew Fonda | eNotes Employee

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Recall that the formula for choosing k people from a group of n people is

`((n),(k)) = (n!) / (k!(n-k)!)`

The answer to your question follows directly from this formula. In both cases, the n=12. In the first case, k=4, so the denominator is 4!*8!. In the second case, k=8, so the denominator is 8!*4!.

This explains why it's true by using the formula, but let's try to develop a more intuitive understanding of why this is true.

Suppose I choose a 4 people from a group of twelve. Since I chose 4 people, this also means that there are 8 people that I did not choose. So in choosing 4 people, I am simultaneously NOT choosing 8 people. So the number of ways I can choose 4 people from 12 is the same as the number of ways I can choose 8 from 12.

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