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

1 question

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

Solution: Treat the 2 specific books as one unit. Then, arrange 6 units: 6! * 2! = 1440.

Steps: 8

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.