Problem 79 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Parking Committee A five-person committee consisting of students and teachers is being formed to study the issue of student parking privileges. Of those who have expressed an interest in serving on the committee, 12 are teachers and 14 are students. In how many ways can the committee be formed if at least one student and one teacher must be included?

Step-by-Step Solution

Answer

The committee can be formed in 62,986 ways with at least one student and one teacher.

Step 1: Calculate Total Committee Formation Ways

First, calculate the total number of ways to form a five-person committee from the pool of 26 individuals (12 teachers and 14 students) without restrictions. Use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where \(n\) is the total number of individuals, and \(r\) is the committee size. Here, \(n = 26\) and \(r = 5\):\[ \binom{26}{5} = \frac{26 \times 25 \times 24 \times 23 \times 22}{5 \times 4 \times 3 \times 2 \times 1} = 65780 \]

Step 2: Committees with Only Students or Only Teachers

Calculate the number of ways to form a committee of only students. Use the combination formula for 14 students:\[ \binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002 \]Similarly, calculate the number of ways to form a committee of only teachers. Use the combination formula for 12 teachers:\[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 \]

Step 3: Apply Inclusion-Exclusion Principle

The number of committees that include at least one teacher and one student can be found by subtracting the invalid scenarios (only students or only teachers) from the total number of combinations. Invalid scenarios:1. All committee members are students: 2002 ways.2. All committee members are teachers: 792 ways.Therefore, the valid committees are:\[ 65780 - 2002 - 792 = 62986 \]

Key Concepts

Counting Principles

Counting principles act as the foundation for many mathematical problems involving selections, arrangements, and group formations. Here, we're dealing with a problem that requires selecting committee members from a group of teachers and students.
To approach such problems, it is crucial to know:

• The Total Number of Choices: Which means understanding how many potential members are available for selection.
• Specific Constraints: Such as needing at least one teacher on the committee.

These principles guide decision-making and ensure that all requirements are met, helping us find the correct number of ways to form the desired group.

Combination Formula

The combination formula plays a critical role in solving problems where the order of selection doesn’t matter. For instance, if determining how many ways a committee can be formed from a group, the order in which members are selected is irrelevant; only their grouping matters.
The formula is given by: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where \(n\) is the total number of people to choose from, and \(r\) is the number of people to choose.
In the exercise, it was crucial to determine the groups of students and teachers separately, utilizing this formula for both. This calculation provides the raw number of possible committee formations before considering additional constraints.

Inclusion-Exclusion Principle

The inclusion-exclusion principle helps when dealing with overlapping sets, or, in simpler terms, when some choices have been erroneously counted twice. In committee formation, this principle helps adjust the raw counts to ensure we don’t include invalid selections.
For example:

• Invalid Scenarios: Such scenarios include selecting all members from one group (e.g., only students or only teachers).
• Correcting Counts: By subtracting the counts of these invalid scenarios, we realign our total to only reflect valid committee structures.

This principle is vital for avoiding those pesky double counts or missed opportunities, and is particularly useful when requirements, like having both students and teachers, come into play.

Committee Formation

Creating a committee with balance and representation is a common task in combinatorics. In this exercise, we formed a committee from 26 individuals - students and teachers, bound by specific rules.

• Basic Structure: The committee needed at least one teacher and one student to ensure diverse representation.
• Dynamic Choices: Different combinations needed to be considered, leading us to utilize various counting tools and subtraction of invalid groups.

Understanding how to tactically form such groups using logic and mathematical formulas can aid in justly distributing roles and making fair representation choices.

Probability

Although not explicitly required in the exercise, understanding probability in this context is beneficial. Probability in selection involves understanding the chances of a particular outcome when forming a group under given constraints.
For the problem:

• Desired Outcome: Forming a committee with at least one teacher and one student out of all possible configurations.
• Total Possible Configurations: Considered by the combination formula outlined previously, providing the base for calculation.

Calculating and understanding the likelihood of valid versus invalid committee formations can be crucial in real-world applications, ensuring fair and desired outcomes in similar selection processes.