Problem 64 - Algebra for JEE ADVANCED

Question

In a group of 13 cricket players, four are bowlers. Find out in how many ways can they form a cricket team of 11 players in which at least 2 bowlers are included. a. 55 b. 72 c. 78 d. None of these

Step-by-Step Solution

Answer

c. 78

Step 1: Total Players and Objective

We have 13 players in total and need to form a team of 11 players. There are 4 bowlers, and we need at least 2 bowlers in the team.

Step 2: Calculate Total Ways to Form a Team of 11

First, determine how many ways we can choose any 11 players out of 13. This can be calculated using the combination formula \( \binom{n}{r} \), where \( n = 13 \) and \( r = 11 \). Therefore, \( \binom{13}{11} = 78 \).

Step 3: Ways with Less Than 2 Bowlers

We calculate the number of ways to form a team with fewer than 2 bowlers (0 or 1 bowler).

Step 4: Ways with 0 Bowlers

Choose 0 bowlers and 11 non-bowlers. Since there are only 9 non-bowlers, forming a team of 11 without any bowlers is impossible, so this contributes 0 ways.

Step 5: Ways with 1 Bowler

Choose 1 bowler and 10 non-bowlers. We have \( \binom{4}{1} \) ways to choose the bowler and \( \binom{9}{10} \) ways for the non-bowlers, but choosing 10 from 9 is impossible. Thus, this also contributes 0 ways.

Step 6: Subtract Invalid Arrangements

Since there are 0 ways to have fewer than 2 bowlers, the valid methods are simply the total combinations with at least 2 bowlers.

Step 7: Final Calculation and Answer

Since only full teams with at least 2 bowlers are valid and all combinations are valid based on previous calculations, the total number of valid teams is 78. Therefore, the answer is 78.

Key Concepts

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, often written as \( \binom{n}{r} \). It represents the number of ways to choose \( r \) elements from a set of \( n \) elements without considering the order. This is essential when determining combinations where arrangement does not matter.

The formula for calculating a binomial coefficient is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n! \) denotes the factorial of \( n \), the product of all positive integers up to \( n \).

Understanding this concept helps in solving problems involving selections or group formations, such as forming teams, choosing committees, and more. In our cricket team problem, it's used to determine how many ways we can choose 11 players from a group of 13.

By calculating \( \binom{13}{11} \), we found there are 78 different ways to select 11 players regardless of their roles.

Combination Formula

The combination formula is a powerful tool used in mathematics to determine the number of ways to select a subset of items from a larger set, where the order of selection does not matter. The formula is represented as \( \binom{n}{r} \).

The significance of the combination formula comes into play when you need to calculate possible selections, like in team or group formation problems. The formula is particularly useful when you must ensure a certain condition is met—in our cricket scenario, it's ensuring at least 2 of the team members are bowlers.

To apply the combination formula, follow these steps:

• Identify the total number of items \( n \) and the subset size \( r \).
• Use the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to calculate the number of combinations.

In the cricket team exercise, after confirming that at least two bowlers are included, the combination formula was valuable in determining there are 78 possible valid team formations.

Team Selection

Selecting a team from a larger pool requires understanding the concepts of permutations and combinations. Team selection is a practical application of combinatorial mathematics, where you choose a specific number of members from a larger group based on certain criteria.

In our example, we're interested in forming an 11-player cricket team from 13 players, ensuring at least 2 bowlers. The emphasis on including a minimum number of bowlers adds a constraint to the problem, which we handle using combination calculations.

When selecting a team, consider:

• The total number of members to choose from and the number of positions available.
• Any constraints, such as specific roles that need to be filled.
• Use of combination formulas like \( \binom{n}{r} \) for calculating the number of arrangements that fit the criteria.

By understanding these factors, students can solve similar problems where specific conditions dictate the selections.

Cricket Team Formation

Forming a cricket team incorporates elements of strategy and mathematics, especially when specific roles or skills must be represented in the team.

In cricket, players usually have specialized skills such as batting, bowling, or wicket-keeping. When forming a cricket team, managers often need to ensure a balanced mix. This is where understanding combinations and team selection comes into play.

Considerations for cricket team formation include:

• The number of players needed for the team.
• Inclusion of a balanced number of specialists, like bowlers and batsmen.
• Using mathematical combinations to ensure all criteria, such as at least 2 bowlers, are met.

In our exercise, establishing a team of 11 with at least 2 bowlers showed how mathematics aids in strategic decision-making. The focus on ensuring specific criteria within a team context is common in sports and provides a practical application for combinatorial principles.