Number of ways to arrange 4 fairy tale books, 5 novels and 3 plays on a shelf when each category stays grouped and the order of categories is fixed

Introduction / Context:
This arrangement problem involves books in three categories that must remain grouped, with the groups themselves arranged in a fixed order. We must count how many ways we can reorder books within each group while preserving the category ordering.

Given Data / Assumptions:

• There are 4 distinct books of fairy tales.
• There are 5 distinct novels.
• There are 3 distinct plays.
• On the shelf, all fairy tales must be placed together, followed by all novels together, followed by all plays together, in that fixed category order.
• Within each category, the order of individual books can vary.

Concept / Approach:
Since the categories must appear in the fixed order fairy tales, then novels, then plays, there is no rearrangement of groups. The only freedom is in arranging the books within each group. Therefore, the total number of arrangements is the product of the permutations within each group.

Step-by-Step Solution:
Step 1: Arrange the 4 distinct fairy tale books among themselves.Step 2: The number of ways to arrange 4 distinct items is 4! = 24.Step 3: Arrange the 5 distinct novels among themselves.Step 4: The number of ways to arrange 5 distinct items is 5! = 120.Step 5: Arrange the 3 distinct plays among themselves.Step 6: The number of ways to arrange 3 distinct items is 3! = 6.Step 7: Since the groups must appear in a fixed order, the total number of arrangements is 4! * 5! * 3!.Step 8: Compute 24 * 120 = 2880 and then 2880 * 6 = 17280.

Verification / Alternative check:
If the category order was not fixed, we would also multiply by 3! for the permutations of the three groups and get 103680. The question specifies the order, so we only consider internal arrangements of each group, confirming that 17280 is correct.

Why Other Options Are Wrong:

• 12400, 17820, and 12460 are similar sized numbers but do not equal 4! * 5! * 3!.
• 17820 is close but not exact, indicating a likely arithmetic error from some partial computation.

Common Pitfalls:
Some students assume that the categories can also be rearranged and therefore multiply by 3!, giving a larger number. Others forget that all books are distinct and undercount by using combinations instead of permutations. Always interpret whether group order is fixed or variable.

Final Answer:
The number of valid shelf arrangements is 17280, so the correct answer is 17280.