Question

There are $10$ pairs of shoes in a cup board from which $4$ shoes are picked at random. The probability that there is at-least one pair is?

• A. $\frac{99}{323}$
• B. $\frac{224}{323}$
• C. $\frac{2}{5}$
• D. $\frac{3}{5}$

Answer 1

Answer: (A)

$\frac{99}{323}$

$10$ pairs$=20$shoes

Probability of atleast $1$ pair$=1-$Probability of no pair

First shoe has $20$ choices

Second shoe has $18$ choices

Third shoe has $16$ choices

Fourth shoe has $14$ choices

Probability of atleast $1$ pair$=1-$Probability of no pair.

$=1-\frac{20}{20}.\frac{18}{19}.\frac{16}{18}.\frac{14}{17}$

$=1-\frac{16\times 14}{19\times 17}$

$=1-\frac{224}{323}=\frac{99}{323}$