In how many ways $11$ players can be selected from $15$ players, if only $6$ of these players can bowl and the $11$ players must include atleast $4$ bowlers?

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Solution

No. of bowlers in team - No. of ways
$4$ $^{6} C_{4}^{9} C_{7}$
$5$ $^{6} C_{5}^{9} C_{6}$
$6$ $^{6} C_{6}^{9} C_{5}$
$∴$ No. of ways is
$^{6} C_{4} .^{9} C_{7} +^{6} C_{5} -^{9} C_{6} +^{6} C_{6} .^{9} C_{5}$
$= 540 + 504 + 126$
$= 1170$

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In how many ways 11 players can be selected from 15 players, if only 6 of these players can bowl and the 11 players must include atleast 4 bowlers?

No. of bowlers in team - No. of ways 4 6C49C7 5 6C59C6 6 6C69C5∴ No. of ways is 6C4.9C7+6C5−9C6+6C6.9C5=540+504+126=1170

In how many ways $11$ players can be selected from $15$ players, if only $6$ of these players can bowl and the $11$ players must include atleast $4$ bowlers?

No. of bowlers in team - No. of ways
$4$ $^{6} C_{4}^{9} C_{7}$
$5$ $^{6} C_{5}^{9} C_{6}$
$6$ $^{6} C_{6}^{9} C_{5}$
$∴$ No. of ways is
$^{6} C_{4} .^{9} C_{7} +^{6} C_{5} -^{9} C_{6} +^{6} C_{6} .^{9} C_{5}$
$= 540 + 504 + 126$
$= 1170$