How many ways can 6 different books be arranged on a shelf if 2 specific books must be together?

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Q: How many ways can 6 different books be arranged on a shelf if 2 specific books must be together?

Solution: Treat the 2 specific books as one unit. Then we have 5 units to arrange: 5! * 2! = 240.

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 6 individual books, we have 5 units to arrange: the 'block' (Book A and Book B together) and the other 4 different books.
Step 3: Calculate the number of ways to arrange these 5 units. The formula for arranging n units is n!. So, we calculate 5! (which is 5 factorial).
Step 4: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step 5: Now, within the 'block', Book A and Book B can be arranged in 2 different ways: either Book A is first and Book B is second, or Book B is first and Book A is second. This is calculated as 2! (which is 2 factorial).
Step 6: Calculate 2! = 2 × 1 = 2.
Step 7: Multiply the number of arrangements of the 5 units by the arrangements of the 2 books in the block: 5! × 2! = 120 × 2 = 240.
Step 8: Therefore, the total number of ways to arrange the 6 different books with the 2 specific books together is 240.