A closet contains $10$ pairs of shoes, If $8$ shoes are selected randomly, what is the probability that there will be exactly 1 complete pair

\frac{1792}{4199}

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\frac{1782}{4199}

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\frac{1772}{4199}

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\frac{1762}{4199}

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Solution

The correct option is A $\frac{1792}{4199}$
Total way to choose 8 shoes from 20 shoes = ${20_{C}}_{8}$
Way to choose one complete pair in 10 pair = ${10_{C}}_{1}$
Way to choose one complete pair in 10 pair on selecting 8 shoes is selecting 6 pairs in remaining 9 and selecting any one by 2 ways= ${9_{C}}_{6}$ $\times$ $2^{6}$ $\times 10$
probability= ${9_{C}}_{6}$ $\times$ $2^{6}$ $\times 10$ / ${20_{C}}_{8}$ =1792/4199

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