Question

A committee of five is to be chosen from a group of $9$people. The probability that a certain married couple will either serve together or not at all is

• A. $\frac{1}{2}$
• B. $\frac{5}{9}$
• C. $\frac{4}{9}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C

$\frac{4}{9}$

An explanation for the correct answer:

Step 1: Find the possible number of ways the committee can be formed.

We have been given that, a committee of five is to be chosen from a group of $9$people.

We need to find the probability that a certain married couple will either serve together or not at all.

Using the formula of combination, possible ways a committee of five is to be chosen from a group of $9$people would be,

$$\begin{gathered} \Rightarrow {}^{9}C_5=\frac{9!}{5!(9-5)!} \\ \Rightarrow {}^{9}C_5=126 \end{gathered}$$

Consider two cases.

Case 1: Couple who serves the committee

Case 2: Couple who does not serve the committee

Step 2: Find the number of couples who serve committee.

$$\begin{gathered} \Rightarrow {}^{7}C_3=\frac{7!}{3!(7-3)!} \\ \Rightarrow {}^{7}C_3=35 \end{gathered}$$

Step 3: Find the number of couples who do not serve committee.

$$\begin{gathered} \Rightarrow {}^{7}C_5=\frac{7!}{5!(7-5)!} \\ \Rightarrow {}^{7}C_5=21 \end{gathered}$$

Step 4: Find the probability that a certain married couple will either serve together or not at all.

The number of married couples who will either serve together or not at all would be,

$21+35=56$

So, the probability that a certain married couple will either serve together or not at all would be,

$$\begin{gathered} \Rightarrow P=\frac{56}{126} \\ \Rightarrow P=\frac{4}{9} \end{gathered}$$

Therefore, option (C) is the correct answer.