Problem 23 - Algebra for College Students

Question

Five associate professors are being considered for promotion to the rank of full professor, but only three will be promoted. How many different combinations of three could be promoted? 10

Step-by-Step Solution

Answer

There are 10 ways to select 3 professors for promotion out of 5.

Step 1: Understanding the Problem

We have 5 associate professors and we need to select 3 for promotion. Since this is a combination problem, we use the combination formula.

Step 2: Recall the Combination Formula

The formula for combinations is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Substitute Values into the Formula

Let's substitute \( n = 5 \) and \( r = 3 \) into the formula: \[ C(5, 3) = \frac{5!}{3!(5-3)!} \]

Step 4: Calculate Factorials

First, calculate the factorials: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \) \( 3! = 3 \times 2 \times 1 = 6 \) \( 2! = 2 \times 1 = 2 \)

Step 5: Apply Factorials to the Formula

Now, apply the calculated factorials into the combination formula: \[ C(5, 3) = \frac{120}{6 \times 2} \]

Step 6: Divide to Find the Result

Simplify and calculate: \[ \frac{120}{12} = 10 \] Thus, there are 10 different combinations of promotions possible.

Key Concepts

Combination Formula

In mathematics, the combination formula is a key concept used in solving problems where the order of selection does not matter. When you have a set of items and need to select a specific number from that set, the combination formula comes into play. It is critical when tackling problems involving selections or groupings.

The combination formula is expressed as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

You use this formula by plugging in the total number of items, \( n \), and the number of selections, \( r \). The formula finds the total number of ways \( r \) items can be chosen from a set of \( n \) items without considering the order.

This is especially useful in cases like our exercise where we have 5 professors and need to choose 3 for promotion. By applying the combination formula, you quickly find the total number of possible groupings.

Factorials

Factorials are a mathematical concept crucial to understanding and applying the combination formula effectively. The factorial of a number \( n \), written as \( n! \), refers to the product of all positive integers up to \( n \).

For example, the factorial of 5, or \( 5! \), is calculated as:

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

Factorials help in determining permutations and combinations where each unique order of selection is important.

To compute the number of combinations using the formula from our exercise, we need the factorials of 5, 3, and 2. Understanding how to calculate factorials allows you to break down complex selection problems into manageable steps, making the problem-solving process more systematic and accessible.

Mathematical Problem Solving

Solving mathematical problems involves a clear understanding and application of concepts and formulas. In combinatorics, recognizing whether a problem involves permutations or combinations is key. With combination problems like the one in our exercise, where order doesn’t matter, simplifying using the combination formula is essential.

To solve these problems efficiently:

• Read and understand the problem context carefully.
• Identify what you are asked to find—in this case, the number of combinations of professors.
• Recall the appropriate formula.
• Substitute known values and calculate.
• Double-check your calculations to ensure accuracy.

By applying these steps, you cultivate critical thinking and problem-solving skills. The exercise highlights the utility of combinations in determining how different groupings can be made without considering the order of selection. This process improves as you practice various types of combinatorial problems.