JEE Main 2025 · 7 April · morning · Single correct · +4 / −1
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) 145
- (b) 165
- (c) 155
- (d) 135
Answer and solution
Answer (c) 155
Solution
Maths sir · 3 steps
Fix the captain and vice-captain first, then fill the other 8 seats case by case.
- Fix the leaders. The captain (a batsman) and the vice-captain (a bowler) must be in the team. That leaves 6 batsmen and 5 bowlers, and 8 more players to choose from them.
- List the splits. The team has 10 players with at least 4 batsmen and at least 4 bowlers. Only 6 bowlers exist, so the possible splits are 4 batsmen + 6 bowlers, 5 + 5, and 6 + 4. In each case, one batsman and one bowler are already fixed.
Batsmen + bowlers Choose more batsmen from 6 Choose more bowlers from 5 Ways 4 + 6 C(6, 3) = 20 C(5, 5) = 1 20 5 + 5 C(6, 4) = 15 C(5, 4) = 5 75 6 + 4 C(6, 5) = 6 C(5, 3) = 10 60 - So. The total is 20 + 75 + 60 = 155 ways, which is option (c).
The idea to keep. When some people must be included, remove them from the pool and from the seats first. Then split by the 'at least' condition and add the cases.
Answer from the NTA final answer key. Solution by PrepWiser.