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 $323p.$

That 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 $1-224/323=\frac{99}{323.}$