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I understand the concept of combinations and permutations. However, I am not getting how to apply the formulas. I believe understanding exactly how to do this would help.

A club has 24 members.
a. In how many ways can 4 officers, a president, vice-president, secretary and treasurer be chosen from the members of the club?
b. In how many ways can a 4-person committee be chosen from the members of the club?

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2 Answers 2

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a. The number is $_{24}P_4 = \dfrac{24!}{20!} = 21\times22\times23\times24 = 255,024$ ways.

b. The number is $_{24}C_4 = \dfrac{24!}{4!\times20!} = 10,626$ ways.

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If you're having a tough time applying the concepts, think of it in pure math. We're picking 4 objects out of 24, with replacement for the first problem. This is a permutation, because there is replacement. so the answer is $_{24} P _4 = 255,024$ ways. For the next one, there clearly can't be replacement, which makes it a combination. Therefore, the answer is $_{24} C _4 = 10,626$ ways.

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