Question

A, B, C, D sit in a row. A is not next to B, and C is next to D. How many arrangements are possible?

• A. 6
  
• B. 12
  
• C. 18
  
• D. 24
  

Hint:

Combine adjacency and non-adjacency restrictions by treating pairs as units and subtracting invalid cases.

Answer 1

The Correct Option is A

- Step 1: Total arrangements. 4 people: $4! = 24$.
- Step 2: C next to D. Treat CD as a unit. Units: (CD), A, B = 3 units. Arrange: $3! = 6$. CD or DC: $2! = 2$. Total = $6 \times 2 = 12$.
- Step 3: A not next to B. In 3-unit arrangement, A and B are adjacent in 2 ways per arrangement. Adjacent = $2 \times 3 = 6$. Non-adjacent = $12 - 6 = 6$.
- Step 4: Verify. Place CD in 3 adjacent pairs (1-2, 2-3, 3-4). CD or DC: $3 \times 2 = 6$. A, B in remaining 2 positions, non-adjacent: 1 way. Total = $6 \times 1 = 6$.
- Step 5: Compare with options. Options: (1) 6, (2) 12, (3) 18, (4) 24. Matches 6.
- Step 6: Conclusion. Option (1) is correct.