Problem 46 - College Algebra

Question

How many ways can a committee of 3 freshmen and 4 juniors be formed from a group of 8 freshmen and 11 juniors?

Step-by-Step Solution

Answer

18,480 ways.

Step 1: Identify the Combination Formula

To solve this problem, we need to use the combination formula, which is used to find the number of ways to choose a subset of items from a larger set. The combination formula is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where \(n\) is the total number of items to choose from, \(k\) is the number of items to choose, and \(!\) denotes a factorial.

Step 2: Calculate Combinations for Freshmen

We have 8 freshmen and need to choose 3. Using the combination formula:\[ C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]Therefore, there are 56 possible ways to choose 3 freshmen from 8.

Step 3: Calculate Combinations for Juniors

We have 11 juniors and need to choose 4. Let's use the combination formula again:\[ C(11, 4) = \frac{11!}{4!(11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330 \]Thus, there are 330 ways to choose 4 juniors from 11.

Step 4: Calculate Total Possible Committees

To find the total number of ways to form a committee of 3 freshmen and 4 juniors, multiply the number of combinations for each group:\[ 56 \times 330 = 18480 \]So, there are 18,480 different ways to form a committee of 3 freshmen and 4 juniors from the given groups.

Key Concepts

Combination Formula

The concept of the combination formula is essential in solving problems involving grouping or selecting items from a larger set. Combinations are a way to determine how many ways we can choose items without regard to the order in which they are selected.

The formula to calculate combinations is:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]where:

• \(n\) is the total number of items.
• \(k\) is the number of items to choose.
• \(!\) denotes factorial, which will be explained next.

This formula allows us to find out how many different groups can be formed. Each group represents a unique selection of items, considering only which items are in the group, not the order of selection. For example, if we have 8 freshmen, and we need to pick 3, we apply the formula to calculate exactly how many different sets of 3 freshmen can be created from the larger group of 8.

Factorial

Factorials are integral to understanding the combination formula as they provide the means to calculate permutations and combinations effectively.The factorial of a number \(n\) (denoted as \(n!\)) is the product of all positive integers up to \(n\).

For example:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• \(3! = 3 \times 2 \times 1 = 6\)

Factorials are useful in combination calculations because they help account for the different ways to arrange items, which is necessary when determining how many unique groups of items can be selected. It's crucial to correctly apply factorials in both the numerator and the denominator in the combination formula, as this balances the calculation by removing the impact of order that's naturally included by the concept of permutations.

Subset Selection

Subset selection is a core concept in combinatorics, involving choosing a smaller set of items from a larger set. In the context of our exercise, subset selection refers to the process of forming a committee from a larger pool of candidates. Selecting subsets involves considering combinations because the arrangement—or order—does not matter.

When you select subsets:

• You first define the total number of items in the original set.
• Then, you decide how many items you need in your subset.

In our committee formation problem, from 8 freshmen, we choose a subset of 3, and from 11 juniors, a subset of 4. These selections are independent of each other, so the possible combinations for each group are calculated separately and then combined (multiplied) to find the total number of ways to form the full committee. This approach highlights the essence of combinatorial counting, focusing on possible groups rather than arrangement within groups.