Question

$8$ 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. $\frac{8}{13}$
• B. $\frac{5}{13}$
• C. $\frac{2}{13}$
• D. $\frac{1}{13}$

Answer 1

Answer: (B)

$\frac{5}{13}$

$10$ pairs of shoes $=20$ shoes
$N(4$ shoes are selected$)=$ ${}^{16}{C}_4$ $=1820$
$P($at least one pair$)=1-P$(no pair$)$
N(no pair) = all $4$ shoes are from distinct pairs
$=$ ${}^8{C}_4\times 2^4$ $=1120$
$P($no pair$)=$ $\frac{N(no.pair)}{N(4.shoes.selected)}$ $=$ $\frac{1120}{1820}$

$P($at least one pair$)=$ $1-\frac{3360}{4845}$
$=\frac{700}{1820}$

$=5/13$