Question

Out of $16$ players of a cricket team, $4$ are bowlers and $2$ are wicket keepers. A team of $11$ players is to be chosen so as to contain at least $3$ bowlers and at least $1$ wicketkeeper. The number of ways in which the team be selected, is

• A. $2400$
• B. $2472$
• C. $2500$
• D. $960$

Answer 1

The correct option is B $2472$

Select 3 bowlers out of 4: $\left(\frac{4}{3}\right)$
(43)
Select 1 wicket keeper out of 2: $\left(\frac{2}{1}\right)$
(21)
Select the remaining 4 players out of 12: $\left(\frac{12}{7}\right)$
(127)

Hence, total permutations $\left(\frac{4}{3}\right)\left(\frac{2}{1}\right)\left(\frac{12}{7}\right)=6336$

So,

Total no of bowlers=4

Total no of wicket keepers=2

Total required selection=

$\left(\frac{4}{3}\right)\left(\frac{2}{1}\right)\left(\frac{10}{7}\right)+\left(\frac{4}{4}\right)\left(\frac{2}{1}\right)\left(\frac{10}{6}\right)+\left(\frac{4}{3}\right)\left(\frac{2}{2}\right)\left(\frac{10}{6}\right)+\left(\frac{4}{4}\right)\left(\frac{2}{2}\right)\left(\frac{10}{5}\right)=2472$