Question

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:

• A. \( 165 \)
• B. \( 155 \)
• C. \( 145 \)
• D. \( 135 \)

Hint:

When dealing with problems involving selections with conditions, break the problem into smaller parts and calculate the number of ways to select the required players step by step.

Answer 1

The Correct Option is B

To form a team of 10 players from 7 batsmen and 6 bowlers, with the constraints of at least 4 batsmen and 4 bowlers, and the captain being a batsman and the vice-captain being a bowler:


Step 1: Select Captain and Vice-Captain
Select 1 batsman from 7 for captain: \( \binom{7}{1} = 7 \) ways.
Select 1 bowler from 6 for vice-captain: \( \binom{6}{1} = 6 \) ways.

Total ways to select captain and vice-captain: \( 7 \times 6 = 42 \).


Step 2: Select Remaining Players
Remaining players to select: 8.
Remaining batsmen needed: At least 3 (to meet the minimum of 4 batsmen, plus captain).
Remaining bowlers needed: At least 3 (to meet the minimum of 4 bowlers, plus vice-captain).

Number of ways to select 3 additional batsmen from the remaining 6: \( \binom{6}{3} \).
Number of ways to select 4 additional bowlers from the remaining 5: \( \binom{5}{4} \).

Total ways to select the remaining 8 players: \( \binom{6}{3} \times \binom{5}{4} = 20 \times 5 = 100 \).


Step 3: Total Selections
Total number of ways to select the team = (Ways to select captain/vice-captain) $\times$ (Ways to select remaining players).

\[42 \times 100 = 4200\]


The total number of ways to select the team is \( 4200 \).