Question

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?

Answer 1

We have, 

Number of players = 17 

Number of players to be select for team =11 

Number of bowlers = 5 

Number of bowlers to be select = 4 

Number of players other than bowlers =12 

4 bowlers out of 5 can be selected in ⁵C₄ ways and 7(i.e., 11-4 = 7) other players can be selected out of 12 (i.e., 17 – 5 = 12) in ¹²C₇ ways. 

∴ Required number of ways = ⁵C₄ x ¹²C₇

= ⁵C₁ x ¹²C₅

∴ ⁿC_r = ⁿC_{n - r}

= 5((12 x 11 x 10 x 9 x 8)/(5 x 4 x 3 x 2 x 1)) = 3960