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Difficulty: MediumBasic Probability and Counting Methods

An art gallery manager is arranging 6 unique paintings side by side in a single line on a display wall. Two specific paintings created by the same artist must be placed directly next to each other. How many different line arrangements of the 6 paintings are possible?

Answer: 240 arrangements

Answer

The total number of different line arrangements is 240.
To satisfy the constraint that two specific paintings must be adjacent, treat them as a single combined block. This reduces the problem to arranging 5 items (the single block plus the other 4 paintings), which can be ordered in 5! = 120 ways. Within the block, the two paintings can be ordered in 2! = 2 ways. Applying the Fundamental Counting Principle gives 120 * 2 = 240 distinct arrangements.

Step-by-Step Solution

1
Group the two specific paintings together as a single block.
There are 5 units to arrange (1 block of 2 paintings plus 4 individual paintings).
Since the two paintings must remain adjacent, treating them as a single item guarantees they will stay together.
2
Calculate the permutations of the 5 units.
5! = 5 * 4 * 3 * 2 * 1 = 120 ways.
There are 5 distinct units being arranged in a straight line.
3
Determine the internal arrangements of the grouped pair.
2! = 2 * 1 = 2 ways.
The two specific paintings within the block can switch positions (Painting A then B, or Painting B then A).
4
Multiply the arrangements together.
120 * 2 = 240 total arrangements.
By the Fundamental Counting Principle, total arrangements equal the number of ways to place the units multiplied by the internal arrangements of the restricted pair.

Key Concept

Permutations with Restrictions (Grouped Elements)
Estimated Time:1m 15s
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Topics

Basic Probability and Counting Methods

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