Question

A students' council with 22 members must select 3 students to help run a fundraiser. This is to be done by randomly selecting 3 names from the 22 names available. The council has 3 students from grade 9, 4 students from grade 10, 10 students from grade 11, and the remainder from grade 12. Determine the probability that 2 out of the 3 names are chosen from the same grade

Answer 1

There are $\dbinom{3}{2}$  ways of selecting  2 names from  grade 9 , and for each of these ways we can choose 1 name from $22-3$ in $\dbinom{19}{1}$ ways.

There are $\dbinom{4}{2}$  ways of selecting  2 names from  grade 10 , and for each of these ways we can choose 1 name from $22-4$ in $\dbinom{18}{1}$ ways.

There are $\dbinom{10}{2}$  ways of selecting  2 names from  grade 11, and for each of these ways we can choose 1 name from $22-10$ in $\dbinom{12}{1}$ ways.

There are $\dbinom{5}{2}$  ways of selecting  2 names from  grade 12, and for each of these ways we can choose 1 name from $22-5$ in $\dbinom{17}{1}$ ways.

The total number of ways to select 2 out of the 3 names from the same grade is the sum

$$\dbinom{3}{2}\dbinom{19}{1}+\dbinom{4}{2}\dbinom{18}{1}+\dbinom{10}{2}\dbinom{12}{1}+\dbinom{5}{2}\dbinom{17}{1}$$

$$=875$$

The total number of ways to select any 3 names from the 22 names available is $\dbinom{22}{3}=1540$

Hence, the probability that 2 out of the 3 names are chosen from the same grade is

$$\dfrac{875}{1540}=\dfrac{25}{44}$$