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Question

Difficulty: EasyPermutations and Linear Arrangements

Four distinct books—two mathematics books and two history books—are to be arranged in a single line on a shelf. If the two mathematics books must stand next to each other, in how many different linear arrangements can the four books be placed?

Answer: 12 arrangements

Answer

12 linear arrangements
To find the total number of linear arrangements where the two mathematics books are adjacent, treat the two mathematics books as a single combined block. This leaves 3 items to arrange (the mathematics block and the 2 individual history books), which can be arranged in 3!=63! = 6 ways. Within the block, the 2 mathematics books can be ordered in 2!=22! = 2 ways. Multiplying the independent arrangements yields 6×2=126 \times 2 = 12 total arrangements.

Step-by-Step Solution

1
Group the adjacent items into a single unit and count total units to arrange
3 units (1 grouped math unit + 2 individual history books), which can be arranged in 3!=63! = 6 ways
Treating items that must be adjacent as a single block simplifies the linear arrangement into distinct available positions.
2
Determine internal permutations within the grouped unit
2!=22! = 2 ways to order the two mathematics books inside their block
The mathematics books can switch positions with each other within their designated block.
3
Multiply external and internal arrangements
6×2=126 \times 2 = 12 total linear arrangements
By the Fundamental Counting Principle, total arrangements equal the product of unit arrangements and internal arrangements.

Key Concept

Linear Permutations with Adjacent Restrictions (Tie-Together Method)
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Topics

Permutations and Linear Arrangements

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