Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Selecting books In how many different ways can five books be selected from a twelve-volume set of books?

Step-by-Step Solution

Answer

There are 792 different ways to select 5 books from a 12-book set.

Step 1: Understanding the Problem

We need to determine the number of different combinations of selecting 5 books out of a set of 12 books. This is a combination problem because the order of selection doesn't matter.

Step 2: Using the Combination Formula

The formula for combinations is given by \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to select. In this scenario, \( n = 12 \) and \( k = 5 \).

Step 3: Substitute Values Into the Formula

Replace \( n \) and \( k \) with 12 and 5, respectively, in the combination formula: \[ C(12, 5) = \frac{12!}{5!(12-5)!} = \frac{12!}{5! \, 7!} \].

Step 4: Simplify the Factorials

Calculate the factorials:\[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7! \]. Notice that \( 7! \) in the numerator and denominator cancel out. Thus,\[ C(12, 5) = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} \].

Step 5: Perform the Division

Calculate the numerator: \( 12 \times 11 = 132 \), \( 132 \times 10 = 1320 \), \( 1320 \times 9 = 11880 \), \( 11880 \times 8 = 95040 \). Now, divide by the denominator: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \), \[ C(12, 5) = \frac{95040}{120} = 792 \].

Key Concepts

Factorials

Factorials are a core part of many mathematical calculations, especially in combinatorics. A factorial is denoted by an exclamation mark (!). It is the product of all positive integers up to a given number. For example, the factorial of 5 is written as 5! and calculated as 5 × 4 × 3 × 2 × 1 = 120.
Factorials are essential in solving combination and permutation problems because they help organize and count different possible arrangements or selections.

• Factorial of 0 and 1 is defined as 1.
• The factorial of any positive integer, n, is the product of all integers from 1 to n.

In the context of combinations, factorials simplify calculations by canceling out elements in the numerator and denominator when using the combination formula. This makes solving complex selection problems feasible and easier to manage.

Combinatorics

Combinatorics is the branch of mathematics exploring ways of selecting objects from a finite set, often without regard to the order of selection. It's a crucial area of study for understanding the principles of permutations and combinations.
In our exercise, we deal specifically with combinations, which are selections where the order is not important.

• Combinatorics focuses on counting and arrangement.
• It helps in determining possible groupings and combinations of elements.
• These techniques are useful in various fields, such as probability, computer science, and optimization problems.

The combination formula, which involves factorials, is a standard part of combinatorics that helps solve numerous practical selection problems.

Selection Problem

Selection problems are a common occurrence in mathematics where one needs to figure out how many possible ways items can be chosen from a larger set. In this instance, we're faced with selecting 5 books from a set of 12.
Selection problems can be addressed either as permutations or combinations, depending on whether order matters.

• If order matters, use a permutation approach.
• If order does not matter, combinations are the solution, as in our exercise.

Understanding whether order is significant dictates the approach (and formula) you will use, critically impacting the calculation and result. This understanding is pivotal as it ensures the problem is solved using the correct methodology, guaranteeing accuracy.

Mathematical Calculations

Mathematical calculations involving combinations require familiarity with the combination formula and its application. For this selection problem, we used the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]This reflects how efficiently mathematical operations allow us to solve selection problems by organizing data into manageable calculations.

• Understand the formula: identify values for n and k based on the problem.
• Substitute these values into the combination formula.
• Simplify the factorial expression to facilitate calculation.

By doing these steps methodically, the complex problem of selecting 5 books from a set of 12 becomes a straightforward calculation of division, leading us to the solution of 792 ways.