Question

Ten pair of shoes are in a closet. Four shoes are selected at random. The probability that there will be at least one pair among the four selected shoes is

• A. $\frac{11}{212}$
• B. $\frac{119}{213}$
• C. $\frac{224}{323}$
• D. $\frac{99}{323}$

Answer 1

The correct option is D $\frac{99}{323}$

$10$ pair means total $20$ shoes.
$p$ (at least one pair is selected) $=1-p$ (no pair)
The total number of ways of selecting four shoes is $20\cdot 19\cdot 18\cdot 17$
We have to choose four shoes in such a way that no pair is selected.
$1^{\text{st}}$ shoe has $20$ choices.
$2^{\text{nd}}$ shoe has $18$ choices.
$3^{\text{rd}}$ shoe has $16$ choices.
$4^{\text{th}}$ shoe has $14$ choices.
$p$ (at least one pair is selected) $=1-\frac{20}{20}\cdot \frac{18}{19}\cdot \frac{16}{18}\cdot \frac{14}{17}=1-\frac{224}{323}$
$p$ (at least one pair is selected) $=\frac{99}{323}$