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9 freshman, 8 sophomores, 6 juniors, and 10 seniors are eligible to be on a committee. if a committee of 14 students is chosen at random, what is the probability that it is made up of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors

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konrad509

[tex] \displaystyle
|\Omega|=\binom{33}{14}=\dfrac{33!}{14!19!}=\dfrac{20\cdot\ldots\cdot33}{2\cdot\ldots\cdot14}=818809200\\
|A|=\binom{9}{2}\cdot\binom{8}{3}\cdot\binom{6}{4}\cdot \binom{10}{5}\\
|A|=\dfrac{9!}{2!7!}\cdot\dfrac{8!}{3!5!}\cdot\dfrac{6!}{4!2!}\cdot\dfrac{10!}{5!5!}\\
|A|=\dfrac{8\cdot9}{2}\cdot\dfrac{6\cdot7\cdot8}{2\cdot3}\cdot \dfrac{5\cdot6}{2}\cdot\dfrac{6\cdot7\cdot8\cdot9\cdot10}{2\cdot3\cdot4\cdot5}\\
|A|=7620480\\\\
P(A)=\dfrac{7620480}{818809200}=\dfrac{10584}{1137235}\approx0.9\% [/tex]

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