Question

There are 10 pairs of shoes in a cupboard from which 4 shoes are picked at random. If p is the probability that there is at least one pair, find 323p

Answer 1

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 no 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 at least one pair is
$1-224/323=99/323$.