Question

Ten pairs of shoes are in a closed. Four shoes are selected at random. The probability that there will be at least one pair among the four shoes selected is

• A. $\frac{90}{323}$
• B. $\frac{99}{323}$
• C. $\frac{47}{323}$
• D. $\frac{54}{323}$

Answer 1

Answer: (B)

$\frac{99}{323}$

$Eachofthe4shoeswechoose,willcomefromonpairs.wecanchoose4pairsin{}^{10}{C}_{4}waysandcanselectashoefromeachpairsin2wayssothatnopairisobtained.Therefore,requiredprobability\Rightarrow 1-\frac{{}^{10}{C}_4\times 2^4}{{}^{20}{C}_4}\Rightarrow 1-\frac{10!}{\begin{array}{c}6!4!\\ \frac{20!}{4!16!}\end{array}}\times 16\Rightarrow 1-\frac{10\times 9\times 8\times 7\times 16}{20\times 19\times 18\times 17}\Rightarrow 1-\frac{224}{323}\Rightarrow \frac{99}{323}$

Hence, this is the answer.