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Difficulty: MediumPermutations, Combinations, and Fundamental Counting Principle

A museum curator is arranging 66 distinct paintings in a single row along a gallery wall. If 22 specific paintings must not be placed adjacent to each other, how many different arrangements of the 66 paintings are possible?

Answer: 480

Answer

480
To find the number of valid arrangements where two specific paintings are not adjacent, use complementary counting. First, compute the total number of ways to arrange 6 distinct paintings without restrictions, which is 6!=7206! = 720. Next, calculate the number of arrangements where the two specific paintings are placed adjacent to one another by treating them as a single block. There are 5 units in total to arrange (the pair block plus the remaining 4 individual paintings), which gives 5!=1205! = 120 ways. Since the two specific paintings can be arranged in 2!=22! = 2 ways inside their block, the total number of adjacent arrangements is 120×2=240120 \times 2 = 240. Finally, subtract the adjacent arrangements from the total arrangements: 720−240=480720 - 240 = 480.

Step-by-Step Solution

1
Calculate the total number of ways to arrange all 6 paintings in a row without any restrictions.
6! = 720
There are 6 distinct items to arrange in 6 sequential positions.
2
Calculate the number of arrangements where the 2 specific paintings are adjacent (placed next to each other).
5! × 2! = 120 × 2 = 240
Treat the 2 specific paintings as a single block unit. This leaves 5 items to arrange (the block + 4 individual paintings), which can be ordered in 5! ways. Within the block, the 2 paintings can be ordered in 2! ways.
3
Subtract the number of adjacent arrangements from the total unrestricted arrangements.
720 - 240 = 480
Complementary counting dictates that valid non-adjacent arrangements equal total possible arrangements minus adjacent arrangements.

Key Concept

Permutations with Adjacency Restrictions (Complementary Counting)
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Topics

Permutations, Combinations, and Fundamental Counting Principle

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