Question

Paula has 10 books that she'd like to read on vacation, but she only has space for 3 books in her suitcase. How many different groups of 3 books can Paula pack?

Short answer

Paula can pack 120 different groups of 3 books.

Step-by-step solution

Understand the Problem

Paula has 10 books and wants to choose 3 out of these 10 books to pack in her suitcase. We need to determine the number of different combinations of 3 books she can make.

Identify the Appropriate Formula

Since the order in which Paula packs the books does not matter, we use the combination formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) where \(n = 10\) and \(k = 3\) in this case.

Plug Values into the Formula

Substitute \(n = 10\) and \(k = 3\) into the combination formula: \( \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \times 7!} \)

Simplify Factorials

Simplify the factorials in the formula: \( 10! = 10 \times 9 \times 8 \times 7! \) Since \(7!\) appears in both the numerator and the denominator, they cancel each other out: \( \frac{10!}{3! \times 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} \)

Perform the Calculations

Calculate the simplified expression: \( \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120 \)

Interpret the Result

The number of different groups of 3 books that Paula can pack is 120.

Key concepts

Combination Formula

In combinatorics, the combination formula is used to determine the number of possible groups or subsets that can be formed from a larger set of items, where the order does not matter. Imagine a scenario where you need to select items but the sequence in which you pick them is irrelevant. This is where the combination formula becomes essential. The formula is expressed as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where:

• is the total number of items, and \(n!\) refers to the factorial of n
• is the number of items you are choosing, and \(k!\) refers to the factorial of k
• The term \(n-k!\) represents the factorial of the difference between the total items and the chosen items.

In Paula's example, she has 10 books and needs to choose 3. Plugging in the values, we get \( \binom{10}{3} = \frac{10!}{3!(10-3)!}\). Understanding and using this formula allows us to find the number of combinations without listing all possibilities.

Factorial Simplification

Factorials play a crucial role in both calculations and simplifications within the combination formula. A factorial, denoted as \( n! \), is the product of all positive integers up to n. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorial simplification makes complex mathematical problems more manageable.
In Paula's problem, with \( n=10 \) and \( k=3 \), the formula gives: $$ \( \binom{10}{3} = \frac{10!}{3! \times 7!}\). Since \(10!\) in the numerator includes \(7!\) as part of its multiplication sequence, we can cancel \(7!\) from both the numerator and the denominator: \(10! = 10 \times 9 \times 8 \times 7!\) and \( 7!\). This yields a simpler product to solve \( \frac{10 \times 9 \times 8}{3 \times 2 \times 1} \). Simplifying factorials is an efficient way to handle large numbers in these calculations.

Problem Solving Steps

Solving combinatorial problems involves a structured approach. Here’s a breakdown:
1. **Understand the Problem**: Clearly identify what's being asked. In this case, Paula has 10 books but can only take 3. We need to find how many unique groups of 3 she can make.
2. **Identify the Appropriate Formula**: Recognize whether you are dealing with permutations or combinations. Since order doesn't matter, use the combination formula \( \binom{n}{k}\).
3. **Plug Values into the Formula**: Substitute your known values into the formula. For Paula, \( n=10 \) and \( k=3 \).
4. **Simplify Factorials**: Break down the factorials to cancel out common terms, making them easier to compute.
5. **Perform the Calculations**: Accurately compute the simplified product to get the result. In this example, \( 10 \times 9 \times 8 = 720\) and \( 3 \times 2 \times 1 = 6\), so \( \frac{720}{6} = 120 \).
6. **Interpret the Result**: Explain what the final number means in the context of the problem. Paula has 120 different groups of 3 books she could pack.

GRE Math Preparation

Preparing for the GRE math section often involves mastering the principles of combinatorics, among other topics. To adeptly tackle problems like Paula's, you should:

• **Familiarize with Formulas**: Knowing when and how to use various formulas like the combination and permutation formulas is crucial.
• **Practice Calculations**: Comfort with simplifying factorials and handling large numbers will save time and reduce mistakes.
• **Understand Concepts Deeply**: Grasp the underlying concepts behind the formulas to apply them effectively in different problems. Ensure you can differentiate between when order matters (permutations) and when it doesn't (combinations).
• **Solve Practice Problems**: Regularly work on practice problems to reinforce these skills and get accustomed to the question formats. Review solutions thoroughly to understand any mistakes.

Combining consistent practice with a deep understanding of core concepts equips you to approach combinatorial problems with confidence on the GRE.