JEE Mathematics 2025 Question with Solution

Given: There are $7$ batsmen and $6$ bowlers. A team of $10$ players is to be selected with at least $4$ batsmen and at least $4$ bowlers. Also, one batsman and one bowler are already designated as captain and vice-captain respectively.

Find: The total number of possible selections.

Since the captain must be a batsman and the vice-captain must be a bowler, first include these two selected players. Then the remaining $8$ players must be chosen from the remaining $6$ batsmen and $5$ bowlers.

Now satisfy the condition of at least $4$ batsmen and at least $4$ bowlers in the final team of $10$.

After already fixing $1$ batsman and $1$ bowler, the remaining $8$ players can be chosen in these valid distributions:

• $3$ batsmen and $5$ bowlers
• $4$ batsmen and $4$ bowlers
• $5$ batsmen and $3$ bowlers

Therefore, the total number of selections is

Evaluating,

So, the total number of ways is $155$. Hence, the correct option is B.

The first approach shown in the source reaches $4200$, which counts captain and vice-captain choices separately and is inconsistent with the final listed option. The correct counting here treats the designated captain batsman and vice-captain bowler as fixed included players, matching the final answer $155$ from the solution.

Given: One batsman and one bowler are already included as captain and vice-captain.

Find: The number of ways to complete the remaining team.

Only the composition of the remaining $8$ players matters. From the remaining $6$ batsmen and $5$ bowlers, choose players so that the final team has at least $4$ batsmen and at least $4$ bowlers.

This gives only three cases:

1. $4$ batsmen and $6$ bowlers in the team $\Rightarrow$ choose $3$ more batsmen and $5$ more bowlers
2. $5$ batsmen and $5$ bowlers in the team $\Rightarrow$ choose $4$ more batsmen and $4$ more bowlers
3. $6$ batsmen and $4$ bowlers in the team $\Rightarrow$ choose $5$ more batsmen and $3$ more bowlers

So,

Therefore, the correct option is B.