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There are 6 batsmen, 7 bowlers and 2 wickets keepers. Find the number of ways of forming a team of 11 players having at least 4 batsmen, at least 5 bowlers and at least 1 wicket keeper.

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Team Size =11 players

Case 1: 4 batsmen and 5 bowlers and 2 wicket-keepers

⇒ 6C× 7C5 × 2C2

= (6×5/2) × (7×6/2) × 1

= 315

Case 2: 4 batsmen and 6 bowlers and 1 wickets keeper

⇒ 6C× 7C6 × 2C1

= (6×5/2) ×7×2

= 210

Case 3: 5 batsmen and 5 bowlers and 1 wickets keeper

⇒ 6C× 7C5 × 2C1

= 6×(7×6/2) ×2

= 252

So total number of ways = 315 + 210 + 252

= 777

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