In how many ways can 6 different books be arranged on a shelf if 2 specific books must be together?
Practice Questions
1 question
Q1
In how many ways can 6 different books be arranged on a shelf if 2 specific books must be together?
120
720
240
60
Treat the 2 specific books as one unit. Then, we have 5 units to arrange: 5! = 120. The 2 books can be arranged in 2! = 2 ways. Total = 120 * 2 = 240.
Questions & Step-by-step Solutions
1 item
Q
Q: In 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! = 120. The 2 books can be arranged in 2! = 2 ways. Total = 120 * 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 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! (5 factorial).
Step 4: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120. This is the number of ways to arrange the 5 units.
Step 5: Now, consider the arrangement of the 2 specific books within their 'block'. Book A and Book B can be arranged in 2 ways: (A, B) or (B, A). This is calculated as 2! (2 factorial).
Step 6: Calculate 2! = 2 × 1 = 2. This is the number of ways to arrange the 2 specific books.
Step 7: To find the total number of arrangements, multiply the number of arrangements of the 5 units by the arrangements of the 2 specific books: 120 (from Step 4) × 2 (from Step 6) = 240.
Step 8: Therefore, the total number of ways to arrange the 6 different books with the 2 specific books together is 240.
