Question

There are 10 pairs of shoes in a cupboards, from which 4 shoes are picked at random. The probability that there is at least one pair, is..........

• A. $\frac{101}{323}$
• B. $\frac{100}{323}$
• C. $\frac{99}{323}$
• D. $\frac{110}{323}$

Answer 1

Answer: (C)

$\frac{99}{323}$

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 pair is selected is $20\times 18\times 16\times 14$
Thus the probability of not getting a pair is $\frac{20\times 18\times 16\times 14}{20\times 19\times 18\times 17}=\frac{224}{323}$
Hence, the probability of getting atleast one pair is $1-\frac{224}{323}=\frac{99}{323}$