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Step-by-Step Solution
Step 1: Understand the problem
We have a squad of 15 players: 6 bowlers, 7 batsmen, and 2 wicketkeepers. We need to form a team of 11 players that includes at least 4 bowlers, 5 batsmen, and 1 wicketkeeper.
Step 2: Identify possible cases
Because we need at least 4 bowlers, 5 batsmen, and 1 wicketkeeper, let us categorize by how many bowlers are chosen:
Case A: 4 bowlers
Case B: 5 bowlers
(Choosing 6 bowlers or more would make it impossible to have at least 5 batsmen and 1 wicketkeeper because the team size is 11.)
Step 3: Calculate the number of ways for each case
Case A: Exactly 4 Bowlers
With 4 bowlers chosen out of 6, the combinations are ^6C_4 .
We have 7 more players to select (since total is 11). They must include at least 5 batsmen and 1 wicketkeeper. The possibilities split further:
Subcase A1: 6 batsmen (out of 7) + 1 wicketkeeper (out of 2)
Subcase A2: 5 batsmen (out of 7) + 2 wicketkeepers (out of 2)
Therefore, combinations for Case A:
^6C_4 \times
\bigl( \, {^7C_6} \times {^2C_1} \;+\; {^7C_5} \times {^2C_2} \bigr).
Case B: Exactly 5 Bowlers
With 5 bowlers chosen out of 6, the combinations are ^6C_5 .
Now we have 6 remaining spots out of 11 (since 11 − 5 = 6). According to the constraints, these 6 must include at least 5 batsmen and 1 wicketkeeper (making exactly 5 batsmen and 1 wicketkeeper). Hence:
{^7C_5} \times {^2C_1}.
Therefore, combinations for Case B:
^6C_5 \times {^7C_5} \times {^2C_1}.
Step 4: Sum the results
Total number of ways =
^6C_4 \Bigl( ^7C_6 \times ^2C_1 \;+\; ^7C_5 \times ^2C_2 \Bigr)
\;+\;
^6C_5 \times {^7C_5} \times {^2C_1}.
Step 5: Numerical Value
First, evaluate the binomial coefficients:
^6C_4 = 15
^7C_6 = 7
^2C_1 = 2
^7C_5 = 21
^2C_2 = 1
^6C_5 = 6
Now calculate:
Inside the parentheses for Case A:
(\,^7C_6 \times ^2C_1 + ^7C_5 \times ^2C_2\,) = 7 \times 2 + 21 \times 1 = 14 + 21 = 35.
Multiply by ^6C_4 = 15 :
15 \times 35 = 525.
For Case B:
^6C_5 \times {^7C_5} \times {^2C_1} = 6 \times 21 \times 2 = 252.
Total:
525 + 252 = 777.
Final Answer
The number of ways to select the team is 777.
