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 $1$ wicket keeper. 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 $2472$

16 players out of whom 4 are bowlers and 2 are wicket-keepers.

A team of 11 players is to be chosen at least 3 bowlers and at least 1 wicket keeper.

Case(1): 3 bowlers and 1 wicketkeeper and remaining 7 players from 10 players.

So, number of ways = ${}_3^4{C.}_1^2{C.}_7^{10}C$

= $\mathrm{4.2.120}=960$

Case(2): 3 bowlers and 2 wicketkeepers.

number of ways = ${}_3^4{C.}_2^2{C.}_6^{10}C$

$=840$

Case(3): 4 bowlers and 1 wicketkeeper.

number of ways = ${}_4^4{C.}_2^1{C.}_6^{10}C$

$=420$

Case(4): 4 bowlers and 2 wicketkeepers.

number of ways = ${}_4^4{C.}_2^2{C.}_5^{10}C$

$=252$

$\therefore$ Number of ways = 960+840+420 = 2472.