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Question

Difficulty: MediumPermutations

Four different Mathematics textbooks and three different Physics textbooks are to be arranged in a line on a shelf. In how many distinct ways can the books be arranged if all four Mathematics textbooks must be kept together?

  1. A
    144
  2. 576Answer
  3. C
    840
  4. D
    5,040

Answer

576 distinct ways
Treating the four Mathematics textbooks as a single unit gives 4 items to arrange on the shelf (the Mathematics unit and the three individual Physics textbooks). These 4 items can be arranged in 4!=244! = 24 ways. Furthermore, the four distinct Mathematics textbooks within the unit can be arranged among themselves in 4!=244! = 24 ways. By the multiplication principle of counting, the total number of distinct arrangements is 24×24=57624 \times 24 = 576.

Step-by-Step Solution

1
Group the Mathematics textbooks into a single block
1 Mathematics block + 3 individual Physics textbooks = 4 items to arrange.
Because all four Mathematics textbooks must remain together, they act as a single composite unit.
2
Calculate the arrangements of the 4 main items
4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 24 ways.
There are 4 distinct items (the Mathematics block and 3 separate Physics books) to place in linear order.
3
Calculate internal permutations of the Mathematics textbooks inside their block
4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 24 ways.
The 4 Mathematics textbooks can be arranged in different orders among themselves.
4
Apply the fundamental counting principle
Total arrangements = 24×24=57624 \times 24 = 576.
Multiply the number of block arrangements by the internal arrangements of the Mathematics textbooks.

Key Concept

Permutations with Restricted Grouping (Block Method)
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Topics

Permutations

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