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Question

Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

A librarian is arranging 77 distinct books on a display shelf: 44 history books and 33 science books. If all 33 science books must be placed adjacent to one another, in how many different linear orders can the 77 books be arranged?

  1. 720Answer
  2. B
    144
  3. C
    5,040
  4. D
    120
  5. E
    35

Answer

720 linear arrangements
To arrange items with an adjacency restriction, bundle the restricted items (the 3 science books) into 1 block. This leaves 4 history books and 1 science block, totaling 5 units to arrange linearly in 5! = 120 ways. Within the science block, the 3 distinct books can be arranged in 3! = 6 ways. By the Fundamental Counting Principle, the total number of linear arrangements is 5! × 3! = 120 × 6 = 720.

Step-by-Step Solution

1
Group the constrained items into a single block.
Treat the 33 distinct science books as a single unit or block, denoted as [S][S].
Since the science books must always stand next to each other, they move as a single combined item.
2
Count the total number of items to arrange linearly.
We have 44 individual history books plus 11 science block, giving 4+1=54 + 1 = 5 items.
The 55 units can be ordered among themselves in 5!5! ways.
3
Calculate the outer arrangements of the 5 units.
5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120.
This accounts for all position permutations of the history books and the block.
4
Calculate internal permutations within the science block and multiply.
3!=63! = 6 ways within the block. Total arrangements = 120×6=720120 \times 6 = 720.
By the Fundamental Counting Principle, total ways equals outer permutations multiplied by inner permutations.

Key Concept

Permutations with Adjacency Restrictions (Block Method)
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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