From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman 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:

Step 1: Identify fixed selections

The problem states that one specific batsman (captain) and one specific bowler (vice-captain) must be included in the team. This means these two positions are already filled, and we need to select the remaining players from the reduced pool.

Step 2: Determine remaining players to select

Since 1 batsman and 1 bowler are already selected, we are left with $7-1=6$ batsmen and $6-1=5$ bowlers to choose from. We need to select $10-2=8$ more players to complete the team of 10.

Step 3: Analyze possible combinations for remaining players

The team must have at least 4 batsmen and at least 4 bowlers. Since one batsman and one bowler are already selected, we need to choose at least 3 more batsmen (total 4) and at least 3 more bowlers (total 4). The total number of players to select is 8. This leads to two possible cases for the remaining 8 players: 3 batsmen and 5 bowlers, or 4 batsmen and 4 bowlers.

Step 4: Calculate combinations for Case 1

In Case 1, we need to select 3 more batsmen from the remaining 6 batsmen and 5 more bowlers from the remaining 5 bowlers. We use the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$.

Step 5: Calculate combinations for Case 2

In Case 2, we need to select 4 more batsmen from the remaining 6 batsmen and 4 more bowlers from the remaining 5 bowlers. Again, we use the combination formula.

Step 6: Calculate total number of ways

The total number of ways to select the team is the sum of the ways from Case 1 and Case 2, as these are mutually exclusive scenarios.

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