How many ways can 7 different books be arranged on a shelf if 3 specific books m

How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?

How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?

Step 1: Identify the 3 specific books that must be together. Let's call them A, B, and C.
Step 2: Treat the 3 specific books (A, B, C) as one single unit or block. Now, instead of 7 books, we have 5 units to arrange: the block (A, B, C) and the other 4 individual books.
Step 3: Count the total number of units. We have 5 units: 1 block (A, B, C) + 4 other books = 5 units.
Step 4: Calculate the number of ways to arrange these 5 units. The formula for arranging n units is n!. So, we calculate 5! (5 factorial).
Step 5: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step 6: Now, within the block (A, B, C), the 3 specific books can also be arranged among themselves. The number of ways to arrange 3 books is 3! (3 factorial).
Step 7: Calculate 3! = 3 × 2 × 1 = 6.
Step 8: Multiply the number of arrangements of the 5 units by the arrangements of the 3 books in the block: 5! × 3! = 120 × 6 = 720.
Step 9: Therefore, the total number of ways to arrange the 7 different books with the 3 specific books together is 720.

Permutations – The arrangement of items in a specific order, where the order matters.
Grouping – Combining specific items into a single unit to simplify the arrangement process.