Question

A student committee of 4 is formed by randomly drawing people (without replacement) from a collection of 4 sophomores, 4 juniors and 3 seniors. What is the probability the committee thus formed will have at least one representative from all three groups of students?

Answer 1

Answer: 0.04848

Explanation:

Given : There are 4 sophomores, 4 juniors and 3 seniors.

Total choices = 4+4+3 = 11

Number of people needed to be chosen for committee = 4

Number of ways to select 3 people out of 11 (without replacement)=
`^(11)P_4`

`=(11!)/((11-4)!)\ \ \ \ [^nP_r=(n!)/((n-r)!)]`

`=(11*10*9*8*7!)/(7!)\\\\= 7920`

Number of ways to select at least one representative from all three groups of students =
`^4P_2* ^4P_1* ^3P_1+^4P_1* ^4P_2* ^3P_1+^4P_1* ^4P_1* ^3P_2`

`=(4!)/(2!)*4*3+4(4!)/(2!)*3+4*4*(3!)/(2!)\\\\= 144+144+96=384`

Required probability =
`(384)/(7920)=0.04848`

Hence, the probability the committee thus formed will have at least one representative from all three groups of students = 0.04848