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:
Show Hint
- \( 165 \)
- \( 155 \)
- \( 145 \)
- \( 135 \)
The Correct Option is B
Solution and Explanation
Step 1: Choose the captain and vice-captain
One batsman (captain) can be selected from 7 batsmen in \( \binom{7}{1} = 7 \) ways. One bowler (vice-captain) can be selected from 6 bowlers in \( \binom{6}{1} = 6 \) ways. So, the number of ways to select the captain and vice-captain is: \[ 7 \times 6 = 42 \]
Step 2: Choose the remaining 8 players
After selecting the captain and vice-captain, we need to select 3 more batsmen and 4 more bowlers (since the team requires at least 4 batsmen and 4 bowlers in total).
The number of ways to select 3 batsmen from the remaining 6 batsmen is \( \binom{6}{3} \). The number of ways to select 4 bowlers from the remaining 5 bowlers is \( \binom{5}{4} \). So, the number of ways to select the remaining 8 players is: \[ \binom{6}{3} \times \binom{5}{4} = 20 \times 5 = 100 \]
Step 3: Calculate the total number of selections
The total number of selections is the product of the number of ways to choose the captain and vice-captain, and the number of ways to choose the remaining 8 players: \[ 42 \times 100 = 4200 \]
Thus, the total number of ways to select the team is \( 4200 \), so the correct answer is \( 155 \).
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