Question

On a shelf, 2 books of History, 3 books of Philosophy, and 5 books of Economics are to be arranged in such a way that the books of any subject are kept together. The number of ways this can be done is

• A. \( 10! \)
• B. 30
• C. 1440
• D. 8640

Hint:

When arranging items in groups, treat each group as a block and arrange the blocks first. Then, arrange the items within each block.

Answer 1

The Correct Option is D

Step 1: Treat each subject's books as one single unit, since they must stay together, giving 3 units in total.
Step 2: These 3 units can be placed on the shelf in \( 3! = 6 \) different orders.
Step 3: Within their own unit, the 2 History books can be ordered in \( 2! = 2 \) ways, the 3 Philosophy books in \( 3! = 6 \) ways, and the 5 Economics books in \( 5! = 120 \) ways.
Step 4: Multiply the unit-ordering count by every internal-ordering count: \( 6 \times 2 \times 6 \times 120 \).
\[ \boxed{8640} \]

Answer 2

A different way to picture this is to first decide which segment of the shelf each subject occupies, and then separately count how each subject's own books can be shuffled inside its segment.

1. Assigning shelf segments to subjects: Imagine dividing the shelf into three contiguous stretches, one of length 2, one of length 3, and one of length 5. The three subjects can be assigned to these stretches in \( 3! = 6 \) ways.
2. Filling the History stretch: Once History is assigned its 2-book stretch, its 2 distinct books can be placed in \( 2! = 2 \) ways.
3. Filling the Philosophy stretch: The 3 distinct Philosophy books can fill their assigned stretch in \( 3! = 6 \) ways.
4. Filling the Economics stretch: The 5 distinct Economics books can fill their assigned stretch in \( 5! = 120 \) ways.

Since assigning subjects to stretches and filling each stretch are independent choices, the multiplication principle applies across all steps: \( 3! \times 2! \times 3! \times 5! = 6 \times 2 \times 6 \times 120 = 8640 \).

Therefore, the correct answer is 8640.