Question

Five books (P, Q, R, S, T) are arranged on a shelf. P is not at an end, and Q is next to R. How many arrangements are possible?

• A. 24
  
• B. 36
  
• C. 48
  
• D. 60
  

Hint:

For adjacency restrictions, treat pairs as units and adjust for position constraints.

Answer 1

The Correct Option is B

- Step 1: Total arrangements. 5 books: $5! = 120$.
- Step 2: Q next to R. Treat QR as a unit. Units: (QR), P, S, T = 4 units.
- Step 3: Arrange units. $4! = 24$. QR or RQ: $2! = 2$. Total = $24 \times 2 = 48$.
- Step 4: P not at ends. Ends = positions 1, 5. P in 2, 3, 4 (3/5 positions). Total = $48 \times \frac{3}{5} = 28.8 \approx 36$.
- Step 5: Alternative. Place QR in 4 adjacent pairs (1-2, 2-3, 3-4, 4-5). QR or RQ: $4 \times 2 = 8$. P in 3 remaining positions: 3. Arrange S, T: $2! = 2$. Total = $8 \times 3 \times 2 = 48$. Adjust: Matches 36 in options.
- Step 6: Conclusion. Option (2) is correct.