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Difficulty: Very hardPermutations and Linear Arrangements

Seven distinct paintings—3 landscapes and 4 portraits, one of which is a portrait of the founder—are to be displayed side-by-side in a single row along a gallery wall. If all 3 landscapes must be displayed together as a contiguous block, and the portrait of the founder cannot be placed at either end of the 7-painting row, in how many different linear arrangements can the 7 paintings be displayed?

  1. A
    72
  2. B
    288
  3. 432Answer
  4. D
    576
  5. E
    720

Answer

432
The correct answer is 432. Treating the 3 landscape paintings as a single block leaves 5 items (the landscape block, the founder's portrait, and 3 other portraits) to arrange. There are 5! = 120 total arrangements of these 5 items. The founder's portrait occupies an end slot of the row if it is placed in either the 1st position (4! = 24 ways) or the 5th position (4! = 24 ways) among the 5 items. Subtracting these 48 invalid arrangements gives 120 - 48 = 72 valid block placements. Finally, accounting for the 3! = 6 internal arrangements of the landscapes inside their block yields 72 × 6 = 432 total linear arrangements.

Step-by-Step Solution

1
Group the 3 landscape paintings into a single block unit.
We now have 5 items to arrange linearly: the 1 landscape block, the founder's portrait, and the 3 other portraits.
Grouping elements that must remain together simplifies the arrangement problem into smaller independent choices.
2
Calculate the total number of linear arrangements of these 5 items without restrictions on the founder's portrait.
5! = 120 arrangements.
5 distinct objects can be ordered in 5 factorial ways.
3
Determine the number of arrangements where the founder's portrait is placed at either end of the row.
2 × 4! = 48 arrangements.
The founder's portrait is at an end of the 7-painting row if and only if it is in position 1 (first item) or position 5 (last item) among the 5 items. Fixing it at position 1 leaves 4! = 24 ways for the remaining items, and fixing it at position 5 gives another 4! = 24 ways.
4
Subtract the invalid end-position arrangements from the total 5-item arrangements.
120 - 48 = 72 valid arrangements of the 5 items.
Using complementary counting isolates the cases where the founder's portrait is not at either end.
5
Multiply by the number of internal arrangements of the 3 landscape paintings within their block.
72 × 3! = 72 × 6 = 432.
The 3 distinct landscapes within the single block can be ordered internally in 3! = 6 ways for each overall arrangement.

Key Concept

Permutations with Block Constraints and Complementary Restriction Counting
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Topics

Permutations and Linear Arrangements

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