Question

Eight people sit around a circular table: P, Q, R, S, T, U, V, W.
Q sits second to the right of P.
S is not a neighbor of R.
Only two people sit between T and W.
U sits opposite V.
How many distinct seatings satisfy all conditions?

Hint:

In circular seating, always fix one person first, then apply rigid positional constraints (like “opposite” or “second to the right”) before handling adjacency restrictions.

Answer 1

Correct Answer: 2

Since this is a circular arrangement, fix P in the top position (at position 1) to remove rotational symmetry.
Step 1: Place Q relative to P. 
Q sits second to the right of P. Facing the center, “right” means clockwise. Thus Q must sit at position 3.
Positions so far: 
1 = P, 3 = Q. 
Step 2: Place U and V. 
U sits opposite V. In an 8-seat table, opposite seats differ by 4 positions. Thus the pair (U,V) must be placed in one of 4 opposite-seat pairs: \[ (2,6), (3,7), (4,8), (5,1) \] But Q is already at position 3, P at position 1, so we eliminate pairs using these positions. 
Remaining valid opposite pairs are: \[ (2,6),\ (4,8) \] Each pair can be assigned as either (U at first, V at second) or (V at first, U at second). 
Thus U–V placements produce: \[ 2 \text{ opposite pairs } \times 2 \text{ ways each} = 4 \text{ possibilities.} \] Step 3: Place T and W. 
Exactly two people sit between T and W. This means: \[ T\text{ at seat }x \Rightarrow W\text{ at }x+3 \text{ or }x-3 \] (modulo 8). 
For each U–V placement, we test all possible T positions and check whether W lands on an available seat.
This step eliminates half of the U–V placements and yields 4 valid placements for (T,W) across all configurations. 
Step 4: Place R and S. 
Remaining two empty seats must be assigned to R and S, but S must not be adjacent to R. Each partial seating from Step 3 leaves exactly two seats open. In half of the cases, these two open seats are adjacent → invalid. In the other half, they are not adjacent → valid. 
Thus from the 4 partial seatings above, only 2 final arrangements remain. Final Count: 
\[ \boxed{2} \] Final Answer: \(\boxed{2}\)