Question

From a heap containing $15$ pairs of shoes, $10$ shoes are selected at random. The probability that there is no complete pair in the selected shoes is

• A. $\frac{{}^{30}{C}_{10}-{(}^{15}{C}_{10})2^{10}}{{}^{30}{C}_{10}}$
• B. $\frac{{}^{15}{C}_{10}\left(2^{10}\right)}{{}^{30}{C}_{10}}$
• C. $\frac{{}^{30}{C}_{10}-2^{15}}{{}^{30}{C}_{10}}$
• D. $\frac{{}^{15}{C}_{10}}{{}^{30}{C}_{10}}$

Answer 1

The correct option is B $\frac{{}^{15}{C}_{10}\left(2^{10}\right)}{{}^{30}{C}_{10}}$

Out of the $15$ pairs, we select any $10$ in $C_{10}^{15}$ ways. Out of these $10$ pairs, we need to select any one of the shoes for each pair so that a complete pair is not formed. This can be done in ${\left(C_1^2\right)}^{10}=2^{10}$ ways. So, favorable cases = $C_{10}^{15}\times 2^{10}$ ways.
So, probability = $\frac{C_{10}^{15}\times 2^{10}}{C_{10}^{30}}$