Question

A faculty student committee is to be selected at random from three students and 4 faculty members. The committee is to consist of two persons. Find the probability that the committee will contain:

a. No student

b. One student

c. Two students

Answer 1

The total number of ways to select 2 people to form a committee is

$$\dbinom{3+4}{2}=\dfrac{7!}{2!5!}=21$$

a.

$$P(no\ students)=\dfrac{\dbinom{3}{0}\dbinom{4}{2-0}}{\dbinom{7}{2}}=\dfrac{1\cdot6}{21}=\dfrac{2}{7}$$

b.

$$P(1\ student)=\dfrac{\dbinom{3}{1}\dbinom{4}{2-1}}{\dbinom{7}{2}}=\dfrac{3\cdot4}{21}=\dfrac{4}{7}$$

c.

$$P(2\ students)=\dfrac{\dbinom{3}{2}\dbinom{4}{2-2}}{\dbinom{7}{2}}=\dfrac{3\cdot1}{21}=\dfrac{1}{7}$$