Question

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are packed at random. The probability that there is at least one pair is

• A. $\frac{99}{323}$
• B. $\frac{224}{323}$
• C. $\frac{100}{323}$
• D. None of these

Answer 1

The correct option is A

$\frac{99}{323}$

Explanation of the correct option.

Step 1: Find the probability of not getting a pair.

Given: There are $10$ pairs of shows, i.e. $20$ shoes.

The total number of ways of choosing $4$ shoes (in order) out of $10$ pairs (or $20$ shoes) is $20\times 19\times 18\times 17$.

The number of ways in which no pair is selected is $20\times 18\times 16\times 14$.

Thus, the probability of not getting a pair is

$$\begin{gathered} \frac{20\times 18\times 16\times 14}{20\times 19\times 18\times 17} \\ =\frac{224}{323} \end{gathered}$$

Step 2: Find the probability of getting at least one pair.

Since, $\mathrm{P}\left(\mathrm{A}\right)\text{'}=1-\mathrm{P}\left(\mathrm{A}\right)$

Therefore, the probability of getting at least one pair is

$$\begin{gathered} 1-\frac{224}{323} \\ =\frac{99}{323} \end{gathered}$$

Hence option $\left(\mathrm{A}\right)$ is the correct option.