Question

A bookshelf contains three novels, six books of poetry, and four reference books. In how many ways can these books be arranged so that the books of each type are together?

Short answer

There are 103,680 ways to arrange the books with each type together.

Step-by-step solution

Consider Each Group as a Single Unit

Begin by treating each type of book—novels, poetry, and reference books—as a single unit or block. This means you are trying to arrange 3 blocks: one for novels, one for poetry, and one for reference books.

Arrange the Groups

Determine the number of ways to arrange these 3 blocks. Since there are 3 distinct groups, you can arrange them in \(3!\) ways. Calculating \(3!\), we get: \(3! = 3 \times 2 \times 1 = 6\).

Arrange Books Within Each Group

Now, consider the ways to arrange the books within each group (block). There are 3 novels that can be arranged among themselves in \(3!\) ways, 6 books of poetry in \(6!\) ways, and 4 reference books in \(4!\) ways. Calculate each factorial: \(3! = 6\), \(6! = 720\), and \(4! = 24\).

Combine Arrangements

Multiply the number of arrangements of the groups by the number of ways to arrange the books within each group to get the total number of arrangements: \(3! \times 3! \times 6! \times 4! = 6 \times 6 \times 720 \times 24 = 103680\).

Key concepts

Understanding Factorials

Factorials are a fundamental concept in permutations and combinations. They help us calculate the total number of ways to arrange a set of items. A factorial, denoted by the symbol "!", is the product of all positive integers less than or equal to a given number. For example, the factorial of 3, written as \(3!\), is calculated as follows: \(3 \times 2 \times 1 = 6\).
In permutations, factorials allow us to determine how to arrange a subgroup within a larger set. Consider this exercise on arranging books. To find out how many ways we can arrange books within each category, we calculate factorials based on the number of books in each category.
- For 3 novels: \(3! = 6\)- For 6 poetry books: \(6! = 720\)- For 4 reference books: \(4! = 24\)
Factorials simplify complex counting problems by reducing them to simple multiplication, which is especially helpful in combinatorial arrangements.

Exploring Combinatorial Arrangements

Combinatorial arrangements involve the selection and arrangement of a subset of items from a larger set. This concept is essential in problems where both selection and arrangement matter, like in this bookshelf scenario.
By grouping books by type and considering each type as a block, the problem transitions into a simple combinatorial arrangement. Instead of treating 13 books individually, they are categorized into 3 distinct blocks (novels, poetry, reference books). This simplification makes it easier to calculate how many ways to organize these groups.
We first arrange the 3 book blocks (or groups) using factorial: there are \(3!\) ways. Once in block form, we then arrange books within each group multiplying their factorial arrangements. These are combined, revealing the power of combinatorial thinking to solve what initially seems like a daunting task.

Grouping in Permutations

When dealing with permutations, especially in problems like arranging books, we often encounter the need to group similar items. Grouping helps manage complexity by simplifying large sets into smaller, more manageable units.
In this exercise, we grouped books by type: novels, poetry, and reference books. By treating each type as a singular block, we effectively reduced the permutation problem. This method stems from the idea of treating distinct item clusters as individual items.
Permutations often rightly focus on ordering, but grouping allows breaking down the task. Once the blocks are ordered using factorial methods, we then dive deeper to address intra-block arrangements, simplifying the overall problem.
This grouping technique highlights how permutations can elegantly handle arrangement issues, adding both depth and clarity to combinatorial problems.