Question

A committee of 6 can be formed from a group of 5 sophomores, 7 juniors, and 8 senior. In how many ways can these be done if:

A. there are 2 members from each class on the committees

B. there shall be exactly 3 seniors, 2 juniors, and 1 sophomore on the committee

Answer 1

Let $A$ denote the event ''sophomore is chosen'', $B$ denote the event ''junior is chosen'', $C$ denote the event ''senior is chosen'',

There are $\dbinom{5+7+8}{6}=38760$ possible outcomes.

A.

$$P(2A, 2B, 2C)=\dfrac{\dbinom{5}{2}\dbinom{7}{2}\dbinom{8}{2}}{\dbinom{20}{6}}$$

$$=\dfrac{10(21)(28)}{38760}=\dfrac{49}{323}=0.1517$$

B.

$$P(1A, 2B, 3C)=\dfrac{\dbinom{5}{1}\dbinom{7}{2}\dbinom{8}{3}}{\dbinom{20}{6}}$$

$$=\dfrac{5(21)(56)}{38760}=\dfrac{49}{323}=0.1517$$