Question

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

• A. 155
• B. 160
• C. 165
• D. 170

Answer 1

Answer: (A)

155

Answer 2

The problem requires selecting 10 players with at least 4 batsmen and 4 bowlers, including a specific captain (batsman) and vice-captain (bowler). Since the captain and vice-captain are fixed, we need to select 8 more players from the remaining 6 batsmen and 5 bowlers. The constraints translate to selecting at least 3 additional batsmen and at least 3 additional bowlers. We consider all valid combinations of selecting these additional players: (3 batsmen, 5 bowlers), (4 batsmen, 4 bowlers), and (5 batsmen, 3 bowlers). The total number of ways is the sum of combinations for these cases: $C(6,3)C(5,5) + C(6,4)C(5,4) + C(6,5)C(5,3) = 20 + 75 + 60 = 155$.