Problem 49 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A committee of 11 members is to be formed from 8 males and 5 females. If \(\mathrm{m}\) is the number of ways the committee is formed with at least 6 males and \(n\) is the number of ways the committee is formed with at least 3 females, then: (a) \(\mathrm{m}+\mathrm{n}=68\) (b) \(\mathrm{m}=\mathrm{n}=78\) (c) \(\mathrm{n}=\mathrm{m}-8\) (d) \(\mathrm{m}=\mathrm{n}=68\)

Step-by-Step Solution

Answer

(b) \( m = n = 78 \)

Step 1: Calculate committees with at least 6 males

First, consider the combinations of forming a committee with at least 6 males. We choose 6, 7, or 8 males from a group of 8. The remaining members to make a total of 11 will be chosen from the 5 females.- **6 males:** Choose 6 males from 8: \( \binom{8}{6} = 28 \) and 5 females from 5: \( \binom{5}{5} = 1 \). So, the total number of ways is \( 28 \times 1 = 28.\)- **7 males:** Choose 7 males from 8: \( \binom{8}{7} = 8 \) and 4 females from 5: \( \binom{5}{4} = 5 \). So, the total number of ways is \( 8 \times 5 = 40.\)- **8 males:** Choose 8 males from 8: \( \binom{8}{8} = 1 \) and 3 females from 5: \( \binom{5}{3} = 10 \). So, the total number of ways is \(1 \times 10 = 10.\)Summing these gives the total \( m \):\[ m = 28 + 40 + 10 = 78 \]

Step 2: Calculate committees with at least 3 females

Next, consider the combinations for forming a committee with at least 3 females. We choose 3, 4, or 5 females from a group of 5. The remaining members to make a total of 11 will be chosen from the 8 males.- **3 females:** Choose 3 females from 5: \( \binom{5}{3} = 10 \) and 8 males from 8: \( \binom{8}{8} = 1 \). So, the total number of ways is \( 10 \times 1 = 10.\)- **4 females:** Choose 4 females from 5: \( \binom{5}{4} = 5 \) and 7 males from 8: \( \binom{8}{7} = 8 \). So, the total number of ways is \( 5 \times 8 = 40.\)- **5 females:** Choose 5 females from 5: \( \binom{5}{5} = 1 \) and 6 males from 8: \( \binom{8}{6} = 28 \). So, the total number of ways is \( 1 \times 28 = 28.\)Summing these gives the total \( n \):\[ n = 10 + 40 + 28 = 78 \]

Step 3: Verify the condition between m and n

Given the calculated values, we have \( m = 78 \) and \( n = 78 \). The options provided are to verify \( m + n = 68 \), \( m = n = 78 \), \( n = m - 8 \), and \( m = n = 68 \).Since both \( m \) and \( n \) equal 78, the valid answer is option (b): \( m = n = 78 \).

Key Concepts

Committee formation

Forming a committee involves selecting a specific number of members from a larger group. In our exercise, we want to form an 11-member committee from a total of 13 potential candidates (8 males and 5 females). There are constraints, such as the committee needing at least 6 males or at least 3 females. These conditions shape our approach to calculating the possible combinations.

Committee formation problems often include restrictions like these, leading us to use combinatorial analysis to determine how many different ways we can select members. It's important to understand that order does not matter in these selections, which is why we use combinations rather than permutations.

Combination formula

The combination formula is crucial for solving problems where the order of selection does not matter. It is used to calculate the number of ways to choose a subset of items from a larger set. The formula is given by:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \( n \) is the total number of items to choose from, \( r \) is the number of items to be chosen, and \(!\) denotes factorial, which means multiplying the entire string of descending positive integers from a given number to 1.

For instance, choosing 6 males out of 8 can be calculated as:\[\binom{8}{6} = \frac{8!}{6! \times 2!} = 28\]We use this formula repeatedly to solve different parts of the problem, whether choosing males or females for the committee. It simplifies the process by offering a straightforward way to calculate the number of possible combinations.

Binomial coefficient

Binomial coefficients are specific numbers that give us the number of combinations available. Represented as \( \binom{n}{r} \), these coefficients are the building blocks of the binomial theorem, which finds broader applications beyond committee selection, such as probability and algebra.

In our problem, these coefficients help us find how many ways there are to select certain numbers of males and females. For example, \( \binom{8}{6} = 28 \) tells us there are 28 ways to choose 6 males from 8. These coefficients help us organize and compute the multitude of ways to pick group members without manually counting each possibility.

This systematic approach enables efficient calculations for larger groups, revealing the importance of binomial coefficients in combinatorial mathematics.

Probability theory

Probability theory is a mathematical framework to quantify chance and is often intertwined with combinatorics when considering combinations and permutations over equally likely outcomes. In the context of committee formation, probability can be used to find the likelihood of certain kinds of committees being formed, though our exercise focuses on possibilities rather than probabilities.

When constraints arise, such as choosing at least a certain number of males or females, calculating the number of combinations becomes a matter of evaluating probabilities indirectly. Each combination represents an outcome, and probability quantifies how often such outcomes might occur if a selection were random.

• Overall, it's vital to see how probability provides a deeper understanding of not merely how many combinations exist, but what they might mean in a practical context.

Understanding probability theory's principles can further expand insights into diverse fields such as game theory, risk assessment, and decision-making under uncertainty.