How many ways can a committee of three be chosen from a group of ten people?

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How many ways can a committee of three be chosen from a group of ten people? How many ways are there to choose a president, secretary, and treasurer.

I know that on the first part I have to use the combination formula since order doesn't matter. Then $\frac{n!}{r!(n-r)!} \rightarrow \frac{10!}{3!(10-3)!}$= 120.

The second part requires order meaning that I need to use the permutation formula $\frac{n!}{(n-r)!}$ $\rightarrow$ $\frac{10!}{(10-3)!}$ = 720.

Is my process correct?

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2 Answers

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Yes, your process is correct. However, normally we to write the question as $\binom{10}{3}$ or ${^{10}\mathrm C_3}$ and the second question as ${^{10}\mathrm{P}_3}$ or $10^{\,\underline 3}$ .

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Yes, you are right. The $3! $ in the second part comes due to the fact that the three chosen people can be assigned the three posts in $^3 P_3 =3! =6$ ways. Hope it helps.

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