Question

There are $15$ members in a cricket club out of which $7$ are batsmen, $6$ are bowlers and $2$ are wicket-keepers. A team of $11$ players is to be formed from these $15$ members. If the coach decides to include at least $4$ batsmen and $1$ wicket-keeper in the team, then the number of ways in which the team can be selected is

• A. $402$
• B. $462$
• C. $572$
• D. $532$

Answer 1

The correct option is C $572$

Possible cases for the team :
$4$ batsmen, $1$ wicket-keeper, $6$ bowlers ;
$5$ batsmen, $1$ wicket-keeper, $5$ bowlers ;
$6$ batsmen, $1$ wicket-keeper, $4$ bowlers ;
$7$ batsmen, $1$ wicket-keeper, $3$ bowlers.

Required number of ways
$={}^7{C}_4\cdot {}^2{C}_1\cdot {}^6{C}_6+{}^7{C}_5\cdot {}^2{C}_1\cdot {}^6{C}_5$
$+{}^7{C}_6\cdot {}^2{C}_1\cdot {}^6{C}_4+{}^7{C}_7\cdot {}^2{C}_1\cdot {}^6{C}_3$

$=70+252+210+40$

$=572$