This question asks about the number of ways to select a cricket team with specific conditions. We have a total pool of 17 players, and we need to form a team of 11 players. A key constraint is that the team must include exactly 3 bowlers from the 5 available bowlers.

Understanding the Cricket Team Selection Problem

The problem is a classic example of combination, where the order of selecting players does not matter. We need to select players for two distinct roles: bowlers and non-bowlers (other players). The total number of players in the pool is 17. Out of these, 5 are bowlers, and the remaining players are non-bowlers. The number of non-bowlers is calculated as:

Total players - Number of bowlers = Non-bowlers

$17 - 5 = 12$ non-bowlers.

We need to select a team of 11 players. The condition is that exactly 3 of these 11 players must be bowlers.

Step-by-Step Team Selection Calculation

To select a team of 11 players with exactly 3 bowlers, we can break the selection process into two independent steps:

1. Selecting the bowlers for the team.
2. Selecting the non-bowlers for the team.

Step 1: Select Bowlers for the Team

We need to select exactly 3 bowlers for the team. There are 5 bowlers available in the total pool of 17 players. The number of ways to select 3 bowlers from 5 available bowlers is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items to choose from, and $k$ is the number of items to choose.

Here, $n=5$ (total bowlers) and $k=3$ (bowlers to select).

Number of ways to select 3 bowlers from 5 = $\binom{5}{3}$ ways.

Step 2: Select Non-Bowlers for the Team

The total team size is 11 players. We have already selected 3 bowlers in Step 1. The remaining players for the team must be non-bowlers. The number of non-bowlers needed for the team is:

Total team size - Number of bowlers selected = Number of non-bowlers needed

$11 - 3 = 8$ non-bowlers.

These 8 non-bowlers must be selected from the pool of non-bowlers. We calculated earlier that there are 12 non-bowlers available (17 total players - 5 bowlers).

The number of ways to select 8 non-bowlers from 12 available non-bowlers is given by the combination formula:

Here, $n=12$ (total non-bowlers) and $k=8$ (non-bowlers to select).

Number of ways to select 8 non-bowlers from 12 = $\binom{12}{8}$ ways.

Calculating Total Ways to Select the Team

Since the selection of bowlers and the selection of non-bowlers are independent events, the total number of ways to form the team of 11 players with exactly 3 bowlers is the product of the number of ways for each step. This is based on the multiplication principle of counting.

Total number of ways = (Ways to select 3 bowlers) $\times$ (Ways to select 8 non-bowlers)

Total number of ways = $\binom{5}{3} \times \binom{12}{8}$.

This can also be written in the notation C(n, k):

Total number of ways = C(5, 3) $\times$ C(12, 8).

Let's briefly look at the values:

• $\binom{5}{3} = \frac{5!}{3!2!} = \frac{5 \times 4}{2 \times 1} = 10$
• $\binom{12}{8} = \frac{12!}{8!4!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$

Total ways = $10 \times 495 = 4950$.

The number of ways is the product of C(5, 3) and C(12, 8).

Analyzing the Options

Let's compare our derived result with the given options:

• Option 1: C(17, 11) - This represents selecting any 11 players from 17 without any conditions on bowlers. This is incorrect.
• Option 2: C(12, 8) - This represents selecting 8 players from 12. This is only one part of our calculation (selecting non-bowlers). This is incorrect.
• Option 3: C(17, 5) $\times$ C(5, 3) - C(17, 5) would be selecting 5 players from 17. This structure doesn't match our requirements. This is incorrect.
• Option 4: C(5, 3) $\times$ C(12, 8) - This matches our calculation: selecting 3 bowlers from 5 AND selecting 8 non-bowlers from 12. This is correct.

Revision Table: Combinations Basics

Additional Information: Variations of Team Selection

Team selection problems often involve various conditions. Here are a few common variations related to selecting players:

• Exactly $k$ players of a specific type: As seen in this problem, you select $k$ from the specific type and the rest from the remaining pool.
• At least $k$ players of a specific type: This usually requires calculating the number of ways for exactly $k$, exactly $k+1$, ..., up to the maximum possible number, and summing these ways. Alternatively, sometimes it's easier to subtract the cases with fewer than $k$ players from the total number of ways without the 'at least' constraint.
• At most $k$ players of a specific type: This involves summing the number of ways for exactly 0, exactly 1, ..., up to exactly $k$ players of that type.
• Excluding certain players: If certain players cannot be selected, you reduce the total pool size and the number of players of the specific type accordingly.
• Including certain players: If certain players must be selected, you include them first, reduce the total team size needed, and reduce the pool of players from which to select the remaining members.

Understanding the composition of the total pool (bowlers vs. non-bowlers in this case) and the requirements for the final team composition is crucial for solving these combination problems correctly using the appropriate formulas and principles like the multiplication principle.