8 couples (husband and wife) attend a dance show ‘Nach Baliye’ in a popular TV channel ; A lucky draw in which 4 persons picked up for a prize is held, then the probability that there is at least one couple will be selected is :
$\dfrac{8}{39}$
$\dfrac{15}{39}$
$\dfrac{12}{13}$
$\dfrac{12}{39}$
None of these
Solution
P (selecting atleast one couple) = 1 - P (selecting none of the couples for the prize.)
$=1 - \left( \dfrac{^{16}C_1 \times ^{14}C_1 \times ^{12}C_1 \times ^{10}C_1}{^{16}C_1 \times ^{15}C_1 \times ^{14}C_1 \times ^{13}C_1}\right)$ $= \dfrac{15}{39}$