How many ways can 4 different books be arranged on a shelf if 2 specific books m

How many ways can 4 different books be arranged on a shelf if 2 specific books must be together?

How many ways can 4 different books be arranged on a shelf if 2 specific books must be together?

Correct Answer: 12

Step 1: Identify the 2 specific books that must be together. Let's call them Book A and Book B.
Step 2: Treat Book A and Book B as one single unit or 'block'. Now, instead of 4 separate books, we have 3 units to arrange: the 'block' (Book A and Book B together) and the other 2 books (Book C and Book D).
Step 3: Calculate the number of ways to arrange these 3 units. The formula for arranging n items is n!. Here, we have 3 units, so we calculate 3! (3 factorial).
Step 4: Calculate 3! = 3 × 2 × 1 = 6. This means there are 6 ways to arrange the 3 units.
Step 5: Now, within the 'block' of Book A and Book B, these 2 books can also be arranged in different ways. The number of ways to arrange 2 items is 2!.
Step 6: Calculate 2! = 2 × 1 = 2. This means there are 2 ways to arrange Book A and Book B within their block.
Step 7: Finally, multiply the number of arrangements of the 3 units by the arrangements of the 2 books in the block: 3! × 2! = 6 × 2 = 12.
Step 8: Therefore, there are 12 different ways to arrange the 4 books on the shelf with the condition that the 2 specific books are together.

Permutations with Constraints – This concept involves arranging items with specific conditions, such as grouping certain items together.
Factorial Calculations – Understanding how to calculate permutations using factorials, particularly when treating grouped items as a single unit.