A library display shelf has space for 5 distinct books aligned in a row. The librarian has… — ACT

A library display shelf has space for $5$ distinct books aligned in a row. The librarian has $3$ different science fiction books and $2$ different biography books. How many different row arrangements of these $5$ books are possible if the $2$ biography books must not be placed next to each other?

Answer: 72 arrangements

Answer

72 different row arrangements are possible.

To find the number of arrangements where the

2

biography books are not adjacent, subtract the number of arrangements where they are adjacent (

4! \times 2! = 48

) from the total possible arrangements (

5! = 120

), resulting in

120 - 48 = 72

Step-by-Step Solution

Calculate the total number of unrestricted arrangements of all 5 distinct books.

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

There are

5

distinct books, so there are

5!

total ways to arrange them in a line.

Calculate the number of arrangements in which the 2 biography books are placed next to each other.

4! \times 2! = 24 \times 2 = 48

Treat the

2

biography books as a single combined unit. This leaves

4

units to arrange (

3

science fiction books plus

1

biography unit), which can be ordered in

4! = 24

ways. Inside the unit, the

2

biography books can be arranged in

2! = 2

ways.

Subtract the number of adjacent biography arrangements from the total number of arrangements.

120 - 48 = 72

Subtracting the restricted outcomes (biography books adjacent) from the total possible outcomes gives the number of valid arrangements (biography books separated).

Key Concept

Permutations and Counting Methods with Complementary Counting

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