Question

A six-person committee is to be chosen from 16 university students, (4 from each class-first, second, third, and fourth years). Determine how many committees are possible if: (a) Each class is represented. (b) No class has more than two representatives, and each class has at least one representative.

Short answer

Part (a): 1024 committees. Part (b): 3456 committees.

Step-by-step solution

Understand the requirements

There are 16 students in total, grouped equally into four classes. We need to form a six-person committee with specific constraints for each scenario. - For part (a), ensure each class is represented at least once in the committee. - For part (b), ensure no class has more than two representatives, and each class has at least one representative.

Calculate for Part (a)

To form a committee where each class is represented, first determine how many students must be chosen from each class. We can choose either 1 student from one class and vary others or have 2 classes represented more. Let's try choosing 1 student from three classes and 3 from one class:- Choose 1 student from the first class: \( \binom{4}{1} \)- Choose 1 student from the second class: \( \binom{4}{1} \)- Choose 1 student from the third class: \( \binom{4}{1} \)- Choose 3 students from the fourth class: \( \binom{4}{3} \)The total number of ways is calculated by multiplying these combinations: \[ \binom{4}{1} \times \binom{4}{1} \times \binom{4}{1} \times \binom{4}{3} \]Calculate the value: \(4 \times 4 \times 4 \times 4 = 256\)Multiply this by the number of ways to choose the class that has 3 representatives (4 possibilities):\(4!/(3!\times 1!) = 4\)Thus, we have: \(4 \times 256 = 1024\) total committees.

Evaluate for Part (b)

For part (b), ensure that no class has more than two representatives, and each has at least one. This can happen in a grouping like 2-2-1-1:- Choose 2 students from the first class: \( \binom{4}{2} \)- Choose 2 students from the second class: \( \binom{4}{2} \)- Choose 1 student from the third class: \( \binom{4}{1} \)- Choose 1 student from the fourth class: \( \binom{4}{1} \)The number of ways of selecting them is: \[ \binom{4}{2} \times \binom{4}{2} \times \binom{4}{1} \times \binom{4}{1} \]Calculate these values:\[ 6 \times 6 \times 4 \times 4 = 576 \] ways.We can rearrange the grouping to get different representations such as 2-1-2-1 or 1-2-2-1, each has 4 permutations:\( 4!/(2! \times 2!) = 6 \) such variations, so multiply this too:So there are \(6 \times 576 = 3456\) ways to select these combinations.

Key concepts

Binomial Coefficient

The binomial coefficient is an essential concept in combinatorics, particularly when it comes to counting combinations. It is denoted as \( \binom{n}{k} \), representing the number of ways to choose \( k \) elements from a set of \( n \) elements, regardless of the order. In simpler terms, it helps us find out how many different groups can be formed from a larger group.
\( \binom{n}{k} \) is calculated using the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

where \( n! \) is the factorial of \( n \). The factorial of a number is the product of all positive integers up to that number.
For example, if you have to choose 2 students from a class of 4, you use the binomial coefficient \( \binom{4}{2} = \frac{4!}{2!(4-2)!} = 6 \). This tells us there are 6 different ways to select 2 students from 4. In committee selection and other combinatorial problems, the binomial coefficient is indispensable to calculate possible groupings.

Combinatorial Constraints

Constraints in combinatorics are the rules or limits that must be adhered to when forming combinations or sets. In problems like committee or group selection, constraints dictate the conditions under which selections are made, affecting the calculation of possible combinations.
In our example, there are specific constraints:

• Part (a) requires each class to be represented in the committee.
• Part (b) ensures no class has more than two representatives and each has at least one.

These constraints significantly affect how we apply the binomial coefficient to solve problems. For instance, in part (a), we must use combinations that ensure all classes are represented, such as choosing 1 student from three classes and 3 from one class.
In part (b), limiting the maximum representatives per class creates different grouping schemes, such as 2-2-1-1, and requires careful calculation to ensure no constraint is violated. Understanding and applying these constraints is crucial for accurate combinatorial calculations.

Committee Selection

Committee selection is a common combinatorial problem where a specific number of people are chosen from a larger group to form a committee. The challenge often lies in adhering to specific selection rules or constraints, as seen in the exercise involving students from different class years.
In this exercise, the committee is formed by selecting 6 students from a pool of 16, divided equally among four class years. The selection conditions can vary, such as ensuring representation from each class or setting limits on the number of representatives per class.
To solve these problems, we apply the binomial coefficient to calculate possible combinations while respecting the imposed constraints. Each scenario might require different grouping strategies, and it’s essential to explore all possibilities that satisfy the selection rules. Successfully tackling committee selection challenges provides a practical understanding of how combinatorial principles work in real-world scenarios.

Student Grouping

Student grouping involves organizing students into specific groups or teams based on set criteria or constraints. It is particularly useful in arranging committees or team projects in educational contexts. In our example, students are grouped into committees under defined rules, such as class representation and limits on members from each class.
To achieve effective student grouping, we must carefully determine the number of students taken from each class while respecting the constraints. In part (a), each class must have at least one representative, hence groups are aligned accordingly. In part (b), each class should have no more than two representatives, necessitating different combination approaches.
These groupings illustrate how combinatorial constraints and binomial coefficients are integrated to solve practical problems. Understanding student grouping enhances our ability to apply combinatorial logic to various educational and organizational tasks, ensuring fair and balanced representation among different groups.