Question

From 12 books in how many ways can a selection of 5 be made, when one specified book is always included, (2) when one specified book is always excluded?

Short answer

In conclusion, there are 330 ways to make a selection of 5 books from 12 when one specified book is always included (\(C(11, 4) = 330\)), and there are 462 ways when one specified book is always excluded (\(C(11, 5) = 462\)).

Step-by-step solution

Case 1: One specified book is always included

If we have a specified book that needs to be included, we can think of it as having already selected one of the books. With this book included, we have to choose the remaining 4 books from the 11 available books. We can find the number of combinations using the formula: \(C(n, r) = \frac{n!}{r!(n-r)!}\) Where \(C(n, r)\) is the number of combinations to choose r elements from a total of n items, and n! is the factorial of n. In this case, we have \(n = 11\) and \(r = 4\): \(C(11, 4) = \frac{11!}{4!(11 - 4)!}\)

Calculating the combinations for Case 1

Calculate the factorial values and simplify the expression: \(C(11, 4) = \frac{11!}{4!7!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330\) So, there are 330 ways to select five books from 12, including the specified book.

Case 2: One specified book is always excluded

If the specified book is always excluded, we now only have 11 books to choose from. We need to find the number of combinations to select 5 books from the remaining 11 books: \(C(11, 5) = \frac{11!}{5!(11 - 5)!}\)

Calculating the combinations for Case 2

Calculate the factorial values and simplify the expression: \(C(11, 5) = \frac{11!}{5!6!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462\) So, there are 462 ways to select five books from 12, excluding the specified book. In conclusion, there are 330 ways to make a selection of 5 books from 12 when one specified book is always included, and there are 462 ways when one specified book is always excluded.

Key concepts

Factorial Notation

Factorial notation is a mathematical concept used to describe the product of an integer and all the non-negative integers below it. It's symbolized by an exclamation point (!). For example, the factorial of 5, written as 5!, is calculated as 5 x 4 x 3 x 2 x 1, which equals 120.

When tackling selection problems in mathematics, factorial notation becomes incredibly useful. It simplifies the process of calculating permutations and combinations, which often involve dividing factorials. For instance, to determine how many ways you can arrange a certain number of items, or select a subset from a larger set, factorials are an essential tool.

In our exercise, we encountered factorials in the formula to calculate combinations. The expression involves dividing factorials to find the number of possible selections. When we say 11!, it means multiplying all whole numbers from 11 down to 1. Understanding factorials paves the way for grasping more complex combinatorial concepts.

Permutations and Combinations

Permutations and combinations are foundational concepts in combinatorial analysis, dealing with the counting of arrangements and selections in a set. While permutations focus on the arrangement of items in a specific order, combinations are concerned with the selection of items where the order does not matter.

For permutations, every detail regarding the arrangement and sequence is important. On the other hand, combinations are used when the position of each item is irrelevant; only the presence or absence of an item in the selection is important. The formula for calculating combinations is given as:\[C(n, r) = \frac{n!}{r!(n-r)!}\]In our exercise, we used this combinations formula to find out how many ways we could select 5 books from 12, under certain conditions.

By understanding the difference between permutations and combinations, students can better comprehend which technique to use in a given problem, thereby avoiding common mistakes and making the problem-solving process more efficient.

Combinatorial Analysis

Combinatorial analysis is the field of mathematics that involves counting, arranging, and selecting objects within a set. It's a broad area that encompasses topics like permutations, combinations, graph theory, and more.

Learning combinatorial analysis equips students with the tools to solve problems involving large sets where direct counting isn't practical. Rather than listing out every possibility, which is often impossible for substantial sets, combinatorial analysis provides formulas and theorems that facilitate faster calculations.

In the textbook exercise, we applied principles of combinatorial analysis to determine the number of ways to select books with certain conditions. For instance, combinatorial analysis tells us that including or excluding a specified item affects the total count, changing the variables in our formulas. By mastering combinatorial concepts, we can approach complex selection problems with confidence and clarity.