Question

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?

Answer 1

Among 22 cricket players there are 3 wicketkeepers + 5 bowlers = 8 players and the remaining (22 – 8) = 14 players are batsmen.

To select 11 players for the team such that it includes exactly 1 wicketkeeper and at least 4 bowlers, the possible selections are,

1 wicketkeeper, 4 bowlers and 6 batsmen.

1 wicketkeeper, 5 bowlers and 5 batsmen.

∴ the number of ways in which a team of 11 players can be chosen

= ³C₁ × ⁵C₄ × ¹⁴C₆ + ³C₁ × ⁵C₅ × ¹⁴C₅

= `3 xx (5!)/(4!1!) xx (14!)/(6!8!) + 3 xx 1 xx (14!)/(5!9!)`

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

= (15 × 7 × 13 × 11 × 3) + (3 × 14 × 13 × 11)

= 45045 + 6006

= 51051.