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: \(\boxed{2}\)

Answer 2

Approach (direct build, seat by seat): Fix P's seat to remove rotational duplicates (a rotated copy of a valid seating isn't a new seating), place Q from the clue, then build outward: fix T and W first, then U and V, and see what's left for R and S.

Step 1: Fix P at a reference seat (position 0 of 8 seats numbered 0 to 7 clockwise). Q sits second to the right of P, so Q is fixed at position 2. This leaves 6 seats {1,3,4,5,6,7} for R, S, T, U, V, W.

Step 2: "Only two people sit between T and W" means their seats are exactly 3 apart along the shorter arc. Checking the 6 free seats, there are exactly 8 ordered ways to place (T,W) so this gap-of-3 condition holds.

Step 3: For each such placement of T and W, 4 seats remain. "U opposite V" means their seats are 4 apart. Exactly 2 ordered ways to place (U,V) satisfy this among the 4 leftover seats.

Step 4: The last 2 seats go to R and S. Checking every case, exactly half of the T-W-U-V placements leave R and S in non-adjacent seats (satisfying "S is not a neighbor of R"); the other half leave them adjacent and must be discarded. Whenever the leftover pair is non-adjacent, R and S can still swap between the two seats, giving 2 arrangements.

Step 5: Combine. 8 (T, W placements) × 2 (U, V placements) give 16 partial layouts; exactly half of these (8) leave R, S non-adjacent, and each such layout allows 2 orders of R and S:
\[ 8 \times 2 = 16. \]

Final Answer: 16 distinct seatings.