Question

There are $10$ pairs of shoes in a cupboard, from which $4$ shoes are picked at random. The probability that there is at least one pair, is $11k/323$. Find the value of $k$.

Answer 1

The total number of ways of choosing $4$ shoes(in order) out of $10$ pairs( so 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 $1-\frac{224}{323}=\frac{99}{323}$
Therefore $k=9$