Problem 75 - Algebra and Trigonometry

Question

55–75 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

6,2986 ways to form the committee.

Step 1: Determine Total Ways to Form Committee

To find the total ways to form a five-person committee from 26 people (12 teachers and 14 students), we use the combination formula \( \binom{n}{r} \), where \( n \) is the total number of people to choose from, and \( r \) is the number of people to choose. 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: Calculate Ways to Form Committee Without Students

We calculate how many committees can be formed with only teachers by choosing 5 teachers out of the 12 available. This ensures no students are on the committee. Again, use the combination formula:\[\binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792\]

Step 3: Calculate Ways to Form Committee Without Teachers

Similarly, calculate how many committees can be formed with only students by choosing 5 students out of the 14 available:\[\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002\]

Step 4: Calculate Committees with At Least One Student and One Teacher

To find committees with at least one student and one teacher, subtract the committees with only teachers and only students from the total. This gives the required number of ways where both student(s) and teacher(s) are included:\[65780 - 792 - 2002 = 62986\]

Step 5: Final Calculation of Total Valid Committees

The number of valid committees, ensuring at least one student and one teacher is included, is thus 62986.

Key Concepts

Combination Formula

The combination formula is a fundamental tool in combinatorics used to determine the number of ways to choose a subset of items from a larger set, without regard to the order in which they are selected. Mathematically, it's expressed as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where

• \( n \) is the total number of available items, and
• \( r \) is the number of items to choose.

The symbol \( ! \) represents a factorial, which is the product of all positive integers up to that number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). The combination formula is particularly useful when order doesn’t matter, like forming committees or selecting team members. In our exercise, we used it to compute different ways to form a committee from teachers and students, showing how each possible group can be derived from the overall selection of interested individuals.

Combinations

Combinations are a particular way of selecting items from a larger set, where the order of selection doesn't matter. This concept is widely applicable in situations where you need to select groups, teams, or subsets.

A simple example is choosing toppings for a pizza. If you have five options and choose three, the combination doesn’t care about the order in which you're selecting the toppings. In mathematical terms, it's the number of subsets of a certain size you can make from a larger set.

In our exercise, combinations were crucial in understanding how to form different types of committees—focusing on groups with students and teachers, or excluding one group to see all potential possibilities. By calculating combinations where no teachers or no students were present, we explored all configurations and ensured the final solution included a mixture of participants.

Counting Principles

Counting principles, like the principle of addition and multiplication, aid in solving problems involving the enumeration of possibilities.

In the context of this exercise, these principles helped determine how many ways we can construct a committee where specific conditions are met, such as having at least one student and one teacher.

• Addition Principle: If there are two separate groups, total possibilities are the sum of the possibilities from each group.
• Subtraction Principle: We used subtraction to eliminate unwanted combinations, like committees of only students or only teachers, to find eligible committees.

In our situation, we first calculated the base number of possible committees, then subtracted the configurations that did not meet our criteria to ensure a diverse committee. This combination of counting principles ensures a logical and systematic approach to evaluating all possible outcomes.