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Maximum Value of nCr

In a touring ...

Question

In a touring cricket team there are $16$ players in all including $4$ bowlers and $2$ wicket-keepers, How many teams of $11$ players from these, can be chosen , so as to include three bowlers and one wicket-keeper

650

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720

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750

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800

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Solution

The correct option is D $720$
$\begin{matrix} T h e r e a r e 4 b o w l e r s 2 w i c k e t - k e e p e r s 10 n o n - b o w l e r s n o t w i c k e t k e e p e r s . C a s e A T h e r e a r e e x a c t l y 3 b o w l e r s, e x a c t l y 1 w i c k e t k e e p e r, a n d 7 n o n - b o w l e r s n o n w i c k e t k e e p e r s .^{3} C_{1} \times^{2} C_{1} \times^{10} C_{7} = 960 C a s e B T h e r e a r e e x a c t l y 4 b o w l e r s, e x a c t l y 2 w i c k e t k e e p e r, a n d 6 n o n - b o w l e r s n o n w i c k e t k e e p e r s .^{4} C_{3} \times^{2} C_{2} \times^{10} C_{6} = 840 C a s e C T h e r e a r e e x a c t l y 4 b o w l e r s, e x a c t l y 1 w i c k e t k e e p e r, a n d 6 n o n - b o w l e r s n o n w i c k e t k e e p e r s .^{4} C_{4} \times^{2} C_{1} \times^{10} C_{6} = 420 C a s e D T h e r e a r e e x a c t l y 4 b o w l e r s, e x a c t l y 2 w i c k e t k e e p e r, a n d 5 n o n - b o w l e r s n o n w i c k e t k e e p e r s .^{4} C_{4} \times^{2} C_{2} \times^{10} C_{5} = 252 N o w, T o t a l n u m b e r s o f w a y s = 960 + 840 + 420 + 252 = 2472 H e n c e, t o t a l n u m b e r o f w a y s i s 2472. \end{matrix}$
Hence,option $B$ is the correct answer.

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