Committee Formation Options
There are 30 men and 40 women in the Town Library Club. They wish to form a 7-person steering committee with 3 men and 4 women. In how many ways can they form the committee?
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
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- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
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- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
In this problem, we're really selecting two separate committees. We can choose 3 men from the 30 men total in $$ \binom{30}{3} = \frac{30 \times 29 \times 28}{3 \times 2 \times 1} = 4,\!060 $$ways, and 4 women from the 40 women total in $$ \binom{40}{4} = \frac{40 \times 39 \times 38 \times 37}{4 \times 3 \times 2 \times 1} = 91,\!390 $$ways. Since these two selections are independent (since for each of the $ 4,\!060 $ ways to pick the men, there are $ 91,\!390 $ ways to pick the women), we multiply them together to get the number of ways that we can form the 7-member overall committee: $$ \binom{30}{3}\binom{40}{4} = (4,\!060)(91,\!390) = \boxed{371,\!043,\!400}.$$
Freeflows