There are 13 cricket players, out of which 4 are bowlers. In how many ways can team of 11 be selected from them so as to include at least 3 bowlers?

Answer : There are 4 bowlers in 13 player team. So, maximum we can add 4 bowlers. And we need to include at least 3 bowlers. If we include 3 bowlers then from the remaining 9 [13 – 4 bowlers] players, we need to include 8. The number of ways, 8 players can be selected among 9 is = ⁹C8 = 9 The number of ways, 3 players can be selected among 4 is = ⁴C3 = 4 So, taking 3 bowlers the team can be represented in (9 X 4) = 36 ways. If we include 4 bowlers then from the remaining 9 [13 – 4 bowlers] players, we need to include 7. The number of ways, 7 players can be selected among 9 is = ⁹C7 = 36 The number of ways, 4 players can be selected among 4 is = ⁴C4 = 1 So, taking 4 bowlers the team can be represented in (36 X 1) = 36 ways. Therefore, the total possible ways are = (36 + 36) = 72.