# Of a group of 50 students, 20 are freshmen, 10 are sophomores, 15 are juniors, and 5 are seniors. 5 members must be chosen. How many different committees can there be if there must be exactly one senior and exactly two freshmen on the committee?

Jan 31, 2017

$285000 \text{ Committees}$.

#### Explanation:

$\left({M}_{1}\right) : 1$ Senior Member, out of $5$, can be chosen in $5$ ways.

$\left({M}_{2}\right) : 2$ Freshmen Members, out of $20$, can be chosen in

""_20C_2={(20)(19)}/{(1)(2)}=190 ways.

Note that, so far, $3$ Members for the Committee, comprising of

$5$ members, have been selected.

$\left({M}_{3}\right) :$ Hence, $2$ members are yet to be selected from $25$ individuals

[10"(Sophomores)+"15"(Juniors)]", and, this can be done in

""_25C_2={(25)(24)}/{(1)(2)}=300 ways.

Using the Fundamental Principle of Counting, there can be

$\left(5\right) \left(190\right) \left(300\right) = 285000$ Committees.