Problem 41 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Choosing Books In how many ways can three books be chosen from a group of six?

Step-by-Step Solution

Answer

There are 20 ways to choose 3 books from a group of 6.

Step 1: Understanding the Problem

We are asked to find the number of ways to choose three books out of six. This is a combination problem because the order in which we choose the books doesn't matter.

Step 2: Identify the Formula for Combinations

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: Plug Values into the Formula

Here, \( n = 6 \) and \( r = 3 \). Substitute these values into the combination formula: \[ C(6, 3) = \frac{6!}{3!(6-3)!} \]

Step 4: Calculate Factorials

Calculate the factorials: - \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \) - \( 3! = 3 \times 2 \times 1 = 6 \) - \( (6-3)! = 3! = 6 \)

Step 5: Solve the Combination Formula

Substitute the factorials into the formula: \[ C(6, 3) = \frac{720}{6 \times 6} = \frac{720}{36} = 20 \]

Step 6: Interpret the Result

Thus, there are 20 different ways to choose 3 books from a group of 6 books.

Key Concepts

Understanding Factorials

In mathematics, a factorial is a function that multiplies a number by all the positive integers below it. This function is central in permutations and combinations. It is represented by an exclamation mark "!". For example:

• The factorial of 5, written as 5!, is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Similarly, for 3!, it is \( 3 \times 2 \times 1 = 6 \).

Using factorials allows us to calculate the total number of arrangements, or permutations. However, in combinations, where the order does not matter, factorials are used in a slightly different way to compute how many groups of items can be selected from a larger pool.

Combination Formula and Its Use

The combination formula is a mathematical method used to find the number of ways to select a certain number of items from a larger set, where the order of selection does not matter. The formula is:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to choose.
• \( n! \) is the factorial of \( n \), \( r! \) is the factorial of \( r \), and \( (n-r)! \) is the factorial of the difference.

This formula calculates the total number of combinations (or groups). For instance, if you have a set of six books and want to select three, you plug the numbers into the formula like so: \( C(6, 3) = \frac{6!}{3!\times(6-3)!} \). In this example, the result is 20. This means there are 20 different ways to choose 3 books from those 6.

Strategically Choosing Books

Choosing books is an example of a real-world application of the concept of combinations. When you need to select a subset of items (such as books) from a larger group, combinations are helpful because you do not care about the order. To illustrate, suppose you have six different books and are tasked with choosing three. You need to know how many sets of three you can form. The order in which you choose the books does not matter. Here’s a simplified way of thinking about it:

• Think of each choice as a slot to fill.
• You are filling 3 slots using the 6 available books.
• Utilize the combination formula to determine all the possible non-ordered selections.

By using the combination formula, you acknowledge that selecting book A before book B is the same as selecting book B before book A. This eliminates any confusion about ordering, and focuses on the fundamental principle: how many different groups can you make? This is what makes combinations so powerful in real-world applications like choosing items without worrying about their order.