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In how many ways can one select a cricket team of eleven from 17 players in which 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?
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Verified Answer
Four blowers can be selected from the five blowers and seven players can be selected from 12 players $(17-5=12)$. If $P$ is the number of ways of selecting the cricket team of eleven, then
$\begin{aligned}
&P=C(5,4) \times C(12,7) \\
&=\frac{5 !}{4 !(5-4) !} \times \frac{12 !}{7 !(12-7) !} \\
&=\frac{5 \times 4 !}{1 \times 4 !} \times \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{5 ! \times 7 !}=3960
\end{aligned}$
$\begin{aligned}
&P=C(5,4) \times C(12,7) \\
&=\frac{5 !}{4 !(5-4) !} \times \frac{12 !}{7 !(12-7) !} \\
&=\frac{5 \times 4 !}{1 \times 4 !} \times \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{5 ! \times 7 !}=3960
\end{aligned}$
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