Question

At Johnson Elementary School, there are 24 educational staff personnel consisting of the following: 7 male teachers, 16 female teachers, plus the principal of the school is a woman named Ruth Klein. Include the Principal as part of the entire educational staff of the school. a) At Johnson Elementary School, in how many ways can a 5-person committee be chosen from the entire educational staff ? Show your work. b) At Johnson Elementary School, if a 5-person committee is chosen at random, what is the probability that it contains exactly 3 men and 2 women chosen from the entire educational staff? You must show how you computed your answer. Round the final answer accurately to 4 decimal places.

Answer 1

a. To solve this, we should make a combination

$\begin{gathered} nCr=\frac{n!}{r!(n-r)!} \\ 24P5=\frac{24!}{5!(24-5)!}=42504 \end{gathered}$

5 person committee can be chosen in 5100480 different ways

b. To find this probability (A), we should use the answer of the last question as the total cases, and the favourable cases will be the shown combinations

$P(A)=\frac{7C3\cdot17C2}{42504}$

Calculate the combinations

$\begin{gathered} 7C3=\frac{7!}{3!(7-3)!}=35 \\ 17C2=\frac{17!}{2!(17-2)!}=136 \\ P(A)=\frac{35\cdot136}{42504}=\frac{4760}{42504}=0.1120 \end{gathered}$