In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can

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6.Permutation and Combination

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In how many ways can one select a cricket team of eleven from $17$ players in which only $5$ players can bowl if each cricket team of $11$ must include exactly $4$ bowlers?

$3960$

Solution

Out of $17$ players, $5$ players are bowlers.

A cricket team of $11$ players is to be selected in such a way that there are exactly $4$ bowlers.

$4$ bowlers can be selected in $^{5} C_{4}$ ways and the remaining $7$ players can be selected out of the $12$ players in $^{12} C_{7}$ ways.

Thus, by multiplication principle, required number of ways of selecting cricket team

$=\,^{5} C_{4} \times \,^{12} C_{7}=\frac{5 !}{4 ! 1 !} \times \frac{12 !}{7 ! 5 !}=5 \times \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1}=3960$

Standard 11

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