Question

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

Explanation:

Step1: Use combination formula

The combination formula is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items and $r$ is the number of items to be chosen.

Step2: Calculate number of ways to choose freshmen

We have $n = 5$ freshmen and we need to choose $r=2$. So $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5!}{2!3!}=\frac{5\times4}{2\times 1}=10$.

Step3: Calculate number of ways to choose sophomores

We have $n = 8$ sophomores and we need to choose $r = 3$. So $C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8!}{3!5!}=\frac{8\times7\times6}{3\times2\times1}=56$.

Step4: Calculate number of ways to choose juniors

We have $n = 8$ juniors and we need to choose $r = 4$. So $C(8,4)=\frac{8!}{4!(8 - 4)!}=\frac{8!}{4!4!}=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70$.

Step5: Calculate number of ways to choose seniors

We have $n = 7$ seniors and we need to choose $r = 5$. So $C(7,5)=C(7,2)$ (since $C(n,r)=C(n,n - r)$). Then $C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7!}{2!5!}=\frac{7\times6}{2\times1}=21$.

Step6: Use multiplication - principle

The total number of ways to form the committee is the product of the number of ways of choosing students from each class. So the total number of ways is $10\times56\times70\times21 = 823200$.

Answer:

$823200$