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:
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Step-by-Step Solution
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)=kn!!(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.