Question

$10$ pairs of shoes are there in a closet. Four shoes are selected at random. The probability that there will be at least one pair is

• A. $\frac{224}{343}$
• B. $\frac{10}{{}^{10}{C}_4}$
• C. $\frac{99}{323}$
• D. $\frac{89}{323}$

Answer 1

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

$10$ pairs of shoes = $20$ shoes
$N(4$ shoes are selected$)=$ ${}^{20}{C}_4$=4845
$P($at least one pair$)=$ $1-P($no pair$)$
$N($no pair$)=$ all $4$ shoes are from distinct pairs
= ${}^{10}{C}_4\times 2^4=3360$
$P($no pair$)=$ $\frac{N(no.pair)}{N(4.shoes.selected)}$ = $\frac{3360}{4845}$
$P($at least one pair$)=$ $1-\frac{3360}{4845}$
= $\frac{1485}{4845}$
= $99/323$