How many ways can 7 different books be arranged on a shelf if 2 specific books m

How many ways can 7 different books be arranged on a shelf if 2 specific books must be together?

How many ways can 7 different books be arranged on a shelf if 2 specific books must be together?

Correct Answer: 1440

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 7 individual books, we have 6 units to arrange: the 'block' (Book A and Book B together) and the other 5 books.
Step 3: Calculate the number of ways to arrange these 6 units. The formula for arranging n items is n!. So, we calculate 6! (which is 6 factorial).
Step 4: Calculate 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
Step 5: Now, within the 'block', Book A and Book B can be arranged in 2 different ways: (A, B) or (B, A). This is 2!.
Step 6: Calculate 2! = 2 × 1 = 2.
Step 7: Multiply the number of arrangements of the 6 units by the arrangements of the 2 books in the block: 720 (from Step 4) × 2 (from Step 6) = 1440.
Step 8: Therefore, the total number of ways to arrange the 7 books with the 2 specific books together is 1440.

Permutations – The arrangement of items in a specific order, considering the total number of items and any constraints.
Grouping – Combining specific items into a single unit to simplify the arrangement process.