Question

question 25: a school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. if 7 freshmen, 7 sophomores, 8 juniors, and 8 seniors are eligible to be on the committee, how many ways can the committee be chosen?

Explanation:

Step1: Calculate freshman combinations

Use combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 7$, $k=2$. So $C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7\times6}{2\times1}=21$.

Step2: Calculate sophomore combinations

Here $n = 7$, $k = 3$. Then $C(7,3)=\frac{7!}{3!(7 - 3)!}=\frac{7\times6\times5}{3\times2\times1}=35$.

Step3: Calculate junior combinations

With $n = 8$, $k = 4$. So $C(8,4)=\frac{8!}{4!(8 - 4)!}=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70$.

Step4: Calculate senior combinations

Given $n = 8$, $k = 5$. Then $C(8,5)=C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8\times7\times6}{3\times2\times1}=56$.

Step5: Find total combinations

Multiply all combinations together: $21\times35\times70\times56 = 176400$.

Answer:

176400