Given:
The number of administrators is a = 6.
The number of teachers is t = 37.
The number of staff is s = 4.
The number of employees required for the committee is r = 4.
The objective is to find the probability that all 4 members are teachers.
Explanation:
The total number of members is,
[tex]\begin{gathered} n=a+t+s \\ =6+37+4 \\ =47 \end{gathered}[/tex]The number of ways in selecting 4 members out of 47 will be,
[tex]\begin{gathered} ^{47}C_4=\frac{47!}{(47-4)!4!} \\ =\frac{47\times46\times45\times44\times43!}{43!4\times3\times2\times1} \\ =178365 \end{gathered}[/tex]Now, the number of ways in selecting 4 members from 37 teachers will be,
[tex]\begin{gathered} ^{37}_{}C_4=\frac{37!}{(37-4)!4!} \\ =\frac{37\times36\times35\times34\times33!}{33!4\times3\times2\times1} \\ =66045 \end{gathered}[/tex]To find the probability:
Now, the probability of forming all 4 members as teachers will be,
[tex]\begin{gathered} P=\frac{66045}{178365} \\ =\frac{4403}{11891} \end{gathered}[/tex]Hence, the probability that all 4 members are teachers is 4403/11891.
