There are 3 wicketkeepers and 5 bowlers among 22 cricket players. A team of 11 players is to be selected so that there is exactly one wicketkeeper and at least 4 bowlers in the team. How many different teams can be formed?

#### Solution

Total number of cricket players = 22

Number of wicketkeepers = 3

Number of bowlers = 5

Remaining players = 14

A team of 11 players consisting of exactly one wicketkeeper and at least 4 bowlers can be selected as follows

(I) 1 wicketkeeper, 4 bowlers, 6 players

or (II) 1 wicketkeeper, 5 bowlers, 5 players

Now, number of selections in (I)

= `""^3"C"_1 xx ""^5 "C"_4 xx ""^14"C"_6`

= `3 xx 5 xx (14 xx 13 xx 12 xx 11 xx 10 xx 9)/(6 xx 5 xx 4 xx 3 xx 2 xx 1)`

= `(15 xx 14 xx 13 xx 2 xx 11 xx 3)/4`

= 45045

Number of selections in (II)

= `""^3"C"_1 xx ""^5 "C"_5 xx ""^14"C"_5`

= `3 xx 1 xx (14 xx 13 xx 12 xx 11 xx 10)/(5 xx 4 xx 3 xx 2 xx 1)`

= 6006

By addition principle, total number of ways choosing a team of 11 players

= 45045 + 6006

= 51051.