Problem 50 - Algebra and Trigonometry

Question

The student relations committee of a college consists of 2 administrators, 3 faculty members, and 5 students. Four administrators, 8 faculty members, and 20 students are eligible to serve. How many different committees are possible?

Step-by-Step Solution

Answer

5,211,264 different committees are possible.

Step 1 - Find the number of ways to choose administrators

First, calculate the number of ways to choose 2 administrators from the 4 eligible ones. This can be represented as a combination: \[ C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

Step 2 - Find the number of ways to choose faculty members

Next, calculate the number of ways to choose 3 faculty members from the 8 eligible ones: \[ C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

Step 3 - Find the number of ways to choose students

Now, calculate the number of ways to choose 5 students from the 20 eligible ones: \[ C(20, 5) = \frac{20!}{5!(20-5)!} = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} = 15,504 \]

Step 4 - Calculate total number of different committees

Finally, multiply all the combinations to find the total number of different committees possible: \[ 6 \times 56 \times 15,504 = 5,211,264 \]

Key Concepts

Core Concept: Combinatorics

Combinatorics is a branch of mathematics dealing with counting, both as a means and an end in obtaining results. It includes the study of combinations, which are selections of items from a set, where the order does not matter.
For example, in the problem of choosing committee members, we use combinatorics to find out the number of ways to select different groups of people.
In this problem, we are interested in forming a committee by choosing from a larger set of candidates. The process of doing this involves calculating combinations for administrators, faculty members, and students separately, and then finding the total number of possible committees by multiplying these combinations together.
For each group, administrators, faculty members, and students, we use a specific combinatorial calculation to find the possible selections. This step-by-step approach helps in understanding the essence of combinatorics and its applications.

Core Concept: Factorial

A factorial, denoted by the symbol '!', is the product of all positive integers up to a given number. Factorial is vital in finding combinations because it helps in counting the arrangements of objects.
For instance, the factorial of 4 (written as 4!) is calculated as:
4! = 4 × 3 × 2 × 1 = 24
When you want to find the number of ways to choose 'k' objects from 'n' objects without considering the order, you use the combination formula: \[ C(n, k) = \frac{n!}{k! (n-k)!} \]
In the initial steps provided, we used factorials to calculate the number of combinations for choosing administrators, faculty members, and students. When calculating these combinations, we often encounter factorials in the denominators and numerators. Simplifying these expressions enables us to find the exact number of ways to select members for the committee.

Core Concept: Committee Selection

Committee selection is a practical application of combinatorics. When selecting a committee, you are typically choosing a subset of individuals from a larger group. The original exercise revolves around selecting such a committee.
In our example:

• We need to choose 2 administrators from a pool of 4.
• 3 faculty members from a pool of 8.
• 5 students from a pool of 20.

For each selection, we used the combination formula to calculate the number of ways we can form each subgroup.
This gives us:
\[ C(4, 2) = 6 \] \[ C(8, 3) = 56 \] \[ C(20, 5) = 15,504 \]
Once we find the combinations for each subgroup, we multiply them together to get the total number of possible committees:
\[ 6 \times 56 \times 15,504 = 5,211,264 \]
This final result tells us there are over 5 million different ways to form such a committee under the given conditions. By understanding the selection process and the use of combinations, the solution demonstrates the diverse and extensive range of possible choices.