Question

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? 
 

• A. 24
• B. 36
• C. 48
• D. 60

Hint:

When people must sit together, treat them as a single block; then account for internal arrangements and apply any further restrictions.

Answer 1

The Correct Option is B

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).