In a seating arrangement, 5 people (A, B, C, D, E) sit in a row. A and B must sit together, and C cannot sit at the ends. How many arrangements are possible?
In a seating arrangement, 5 people (A, B, C, D, E) sit in a row. A and B must sit together, and C cannot sit at the ends. How many arrangements are possible?
Show Hint
- 24
- 36
- 48
60
The Correct Option is B
Solution and Explanation
Step 1: Treat A and B as One Unit
This ensures they sit together. Now we have units: (AB), C, D, E → total 4 units.
Step 2: Arrange the Units
Number of arrangements: 4! = 24 ways.
Step 3: Internal Arrangement of A and B
Within (AB), they can be (A, B) or (B, A), so 2 ways.
Without restrictions on C, total arrangements = 24 × 2 = 48.
Step 4: Apply Restriction on C
C cannot be in positions 1 or 5. Number of ways to place C in middle positions = 3 choices.
Step 5: Arrange Remaining People
After placing C, we arrange (AB) as a block + 2 other individuals in remaining 4 seats:
Arrangements = 3! × 2 = 6 × 2 = 12 ways.
Step 6: Total Arrangements
3 choices for C × 12 arrangements = 36.
Step 7: Conclusion
Total arrangements = 36, matching option (2).
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