In a group of $13$ cricket players, four are bowlers. Find out in how many ways can they form cricket team of $11$ players in which at least $2$ bowlers are included.

55

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72

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78

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None of these

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Solution

The correct option is D $78$
No. of bowlers $= 4$
Other $= 13 - 4 = 9$
No. of ways $=^{4} C_{2}^{9} C_{9} +^{4} C_{3}^{9} C_{8} +^{4} C_{4}^{9} C_{7}$
$= \frac{4!}{2! 2!} + \frac{4!}{3!} \frac{9!}{8!} + \frac{9!}{7! 2!}$
$= \frac{24 \times 3}{2} + 4 \times 9 + \frac{9 \times 8}{2}$
$= 6 + 36 + 36$
$= 78$ .

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In a group of 13 cricket players, four are bowlers. Find out in how many ways can they form cricket team of 11 players in which at least 2 bowlers are included.

The correct option is D 78No. of bowlers=4Other=13−4=9No. of ways=4C29C9+4C39C8+4C49C7=4!2!2!+4!3!9!8!+9!7!2!=24×32+4×9+9×82=6+36+36=78.

In a group of $13$ cricket players, four are bowlers. Find out in how many ways can they form cricket team of $11$ players in which at least $2$ bowlers are included.