**Question:**

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?

**Solution:**

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 $={ }^{9} \mathrm{C}_{8}=9$ The number of ways, 3 players can be

selected among 4 is $={ }^{4} \mathrm{C}_{3}=4$ 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 $={ }^{9} \mathrm{C}_{7}=36$ The number of ways, 4 players can be selected among 4 is $={ }^{4} \mathrm{C}_{4}=1$ So,

taking 4 bowlers the team can be represented in $(36 \times 1)=36$ ways. Therefore, the total possible ways are $=(36+36)=72$.