Question

Ten pairs of shoes are in a closet . Four shoes are selected at random . The probability that there will be strictly 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)

$\frac{99}{323}$

Total no. of ways of selecting shoes${}^{20}{\text{C}}_4$

Probability that there will be atleast one pair among the four shoes selected is

let one pair of shoes be one unit ${\therefore }^{10}{\text{C}}_1{\cdot }^{18}{\text{C}}_2$

and same goes for two pairs of shoes ${\therefore }^{10}{\text{C}}_2{\cdot }^{16}{\text{C}}_0$

$\therefore \frac{{}^{10}{\text{C}}_1{\cdot }^{18}{\text{C}}_2{-}^{10}{\text{C}}_2{\cdot }^{16}{\text{C}}_0}{{}^{20}{\text{C}}_4}$

$=\frac{99}{323}$

Answer