Answer
$30,030$ ways
Work Step by Step
Committee members are chosen with no importance of order of choice, so we deal with combinations.
We have a sequence of selections in which we choose
1. ... 2 out of a group of 5 professors ... in ${}_{5}C_{2}$ ways
2. ... 10 out of a group of 15 students... in ${}_{15}C_{10}$ ways
By the Fundamental Counting Principle,
Total ways= ${}_{7}C_{4}\cdot {}_{7}C_{5}$
${}_{5}C_{2}=\displaystyle \frac{5!}{(5-2)!2!}=\frac{5\times 4}{1\times 2}=10$
${}_{15}C_{10}=\displaystyle \frac{15!}{(15-10)!10!}$
$=\displaystyle \frac{15\times 14\times 13\times 12\times 11}{1\times 2\times 3\times 4\times 5}=3003$
Total = $10\times 3003$ = $30,030$ ways
$30,030$ ways
