You have four favourite love story books displayed between two book-ends in your study. You decide to arrange these four books in every possible order, changing the arrangement so that, over time, each distinct ordering of the four books appears once. You move just one book per minute to create a new arrangement. Assuming you can create each of the possible distinct arrangements and that you take one minute per arrangement, how long will it take you to go through all possible different orders of the four books?

Introduction / Context:
This is a permutations question framed as a time puzzle. It asks you to calculate how many different ways four distinct books can be arranged on a shelf and then interpret that number as minutes, given that you take one minute per arrangement.

Given Data / Assumptions:

• There are 4 different books, all distinguishable from one another.

• You are interested in every possible ordering of these 4 books between two book-ends.

• You assume that each distinct arrangement takes 1 minute to set up.

• You will not repeat any arrangement once it has already been used.

Concept / Approach:
The number of different arrangements (permutations) of n distinct objects in a line is n!. Here, n = 4, so the total number of possible distinct arrangements is 4! (4 factorial). After finding this count, we directly convert that number into minutes because each arrangement corresponds to one minute of time.

Step-by-Step Solution:
Step 1: The number of ways to arrange 4 distinct books in a line is 4!. Step 2: Compute 4!: 4! = 4 * 3 * 2 * 1 = 24. Step 3: Therefore, there are 24 distinct orders of the four books. Step 4: Since you take 1 minute to place the books in each of these arrangements, the total time needed is 24 minutes.

Verification / Alternative check:
We can think in a constructive way: choose the first place in 4 ways, the second place in 3 ways (remaining books), the third in 2 ways, and the last in 1 way. Thus the number of arrangements is 4 * 3 * 2 * 1 = 24, which matches 4!. This confirms that 24 is the correct number of distinct orders.

Why Other Options Are Wrong:
Option A (54 minutes), Option B (32 minutes), and Option C (48 minutes) all correspond to numbers of arrangements greater than the maximum possible 24. They would imply more distinct permutations than actually exist for 4 objects, which is impossible.

Common Pitfalls:
A common mistake is to confuse permutations with combinations and think that order does not matter, leading to a much smaller number, or to overcomplicate by imagining constraints that are not given. Another error is multiplying incorrectly when computing factorials. Remember that for n distinct books, every different order counts as a separate arrangement, so permutations are the correct concept to use.

Final Answer:
Hence, it will take you 24 minutes to go through all possible distinct arrangements of the four books.