Problem 56 - Algebra for College Students

Question

How many seven-person committees consisting of three juniors and four seniors can be formed from 45 juniors and 53 seniors? \(4,155,186,750\)

Step-by-Step Solution

Answer

The number of ways to form the committee is 3,546,315,000.

Step 1: Understand the Task

We are asked to form a committee consisting of seven people: three juniors and four seniors. We are given that there are 45 junior students and 53 senior students.

Step 2: Calculate Ways to Choose Juniors

To choose 3 juniors out of 45, we use the combination formula \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order. Thus, calculate \(\binom{45}{3}\), which is \(\frac{45!}{3!(45-3)!}\).

Step 3: Simplify the Calculation for Juniors

Calculate \(\binom{45}{3} = \frac{45 \times 44 \times 43}{3 \times 2 \times 1} = 14,190\). This is the number of ways to choose 3 juniors from 45.

Step 4: Calculate Ways to Choose Seniors

To choose 4 seniors out of 53, again use the combination formula \(\binom{53}{4}\), which is \(\frac{53!}{4!(53-4)!}\).

Step 5: Simplify the Calculation for Seniors

Calculate \(\binom{53}{4} = \frac{53 \times 52 \times 51 \times 50}{4 \times 3 \times 2 \times 1} = 249,900\). This is the number of ways to choose 4 seniors from 53.

Step 6: Combine Juniors and Seniors Selections

Since the committee is formed by selecting both juniors and seniors, multiply the number of ways to select the juniors by the number of ways to select the seniors: \(14,190 \times 249,900\).

Step 7: Calculate Total Combinations

Calculate \(14,190 \times 249,900 = 3,546,315,000\). This is the total number of ways to form the committee.

Key Concepts

Combination Formula

In combinatorics, a combination is a way to select items from a larger pool where order does not matter. This is crucial in solving problems like forming committees or teams by choosing members from a larger group where their order within the team is irrelevant.
To calculate combinations, we use the formula \(\binom{n}{k}\), which denotes the number of ways to choose \(k\) items from \(n\) items without regard to the sequence in which they are selected. The formula is given by:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
This formula uses factorial notation, where \(n!\) (read as "n factorial") means the product of all positive integers up to \(n\).
For instance, choosing 3 juniors out of 45 is calculated as follows:

• \(\binom{45}{3} = \frac{45!}{3! \times 42!} = \frac{45 \times 44 \times 43}{3 \times 2 \times 1} = 14,190\)

This calculation tells us there are 14,190 ways to select 3 juniors without considering order.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics. They help in determining the different ways to arrange or select items from a group. While permutations consider the arrangement order, combinations do not.
Permutations are used when the sequence of selection is important. For example, if organizing people in a line, each different order counts as a separate permutation.
Combinations, on the other hand, are for selecting items where arrangement does not matter, such as forming a committee. The problem of forming committees of certain sizes from a group, as seen in the juniors and seniors example, is an ideal use of combinations.
Let's recap:

• Permutations are used when order matters.
• Combinations are used when order does not matter.

In our committee exercise, the use of combinations simplifies the task by eliminating the concern for order as the committee’s composition is the only aspect of interest.

Mathematical Problem Solving

Mathematical problem-solving in combinatorics involves identifying the structure of the problem and deciding whether permutations or combinations are appropriate. Such exercises refine one's logical thinking and understanding of how combinations apply to real-world scenarios.
To solve combinatorial problems like committee formation, follow these steps:

• Understand the problem's requirements. For example, the number of juniors and seniors needed on the committee.
• Apply the combination formula to each group—calculate the combinations separately for juniors and seniors.
• Combine the results by multiplication to find the total number of ways both groups can be selected. In this instance, multiply the number of ways to choose the juniors and the seniors.

Such methodical approaches improve decision-making skills and hone your ability to solve complex problems efficiently. The logic and clarity in combining the selections lead to the complete solution: for instance, our calculated total for committee forming was 3,546,315,000.