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(A) Total number of ways to select $4$ shoes out of $20$ is given by $\binom{20}{4} = \frac{20 	imes 19 	imes 18 	imes 17}{4 	imes 3 	imes 2 	im

Vedclass · Accepted Answer

$\frac{99}{323}$

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(A) Total number of ways to select $4$ shoes out of $20$ is given by $\binom{20}{4} = \frac{20 	imes 19 	imes 18 	imes 17}{4 	imes 3 	imes 2 	imes 1} = 4845$.To find the probability of at least one pair,we calculate the complement: the probability that no pair is selected.To select $4$ shoes such that no two form a pair,we must first choose $4$ distinct pairs out of $10$,which can be done in $\binom{10}{4}$ ways. From each of these $4$ pairs,we must choose $1$ shoe,which can be done in $2^4$ ways.Number of ways to select $4$ shoes with no pair $= \binom{10}{4} 	imes 2^4 = 210 	imes 16 = 3360$.Probability of no pair $= \frac{3360}{4845} = \frac{224}{323}$.Probability of at least one pair $= 1 - \frac{224}{323} = \frac{99}{323}$.

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

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