AMC 10 · 2008 · #21

Grade 7 counting
Archetype Branching State Enumeration └ Sub-archetype Ordered Placement Count
permutations-basiccasework systematic-enumeration ↑ Prerequisites: permutations-basic
📏 Medium solution 💡 3 insights
Problem
Ten chairs, numbered 1 through 10 clockwise, sit evenly spaced around a round table. Five married couples (five men and five women) take the chairs so that genders alternate around the circle. No person may sit in a chair next to their spouse, and no person may sit directly across the table from their spouse. Count how many seating arrangements obey all of these rules.

Pick an answer.

(A)
$\ 240$
(B)
$\ 360$
(C)
$\ 480$
(D)
$\ 540$
(E)
$\ 720$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The men's rule and the women's rule are tangled together, so tool #7 (Identify Subproblems) splits the count into two clean stages: first seat the five men using only the alternating rule, then, with the men frozen in place, count how many ways the five wives can fill the opposite-gender chairs without landing next to or across from their husbands. Tool #1 (Draw a Diagram) makes the geometry concrete: mark which chairs are neighbors and which are across, so each man's three forbidden chairs are easy to read off. Tool #2 (Make a Systematic List) then shows that every wife has only two legal chairs, and those choices lock together into a single loop, so only two full women-arrangements survive. Multiplying the men-count by the women-count gives the answer.

1STEP 1

Seat the five men

Men take all the odd chairs or all the even chairs — 2 choices — and 5! = 120 orders inside them, so 2 × 120 = 240 ways.

2 × 5! = 2 × 120 = 240
2STEP 2

Find each wife's legal chairs

Fix men in 1,3,5,7,9. Man 1 blocks chairs 2, 10 and 6, leaving his wife only {4,8}; every other wife likewise has exactly two chairs.

man 1→{4,8}, 3→{6,10}, 5→{8,2}, 7→{10,4}, 9→{2,6}
3STEP 3

Chain the wives: only two arrangements

The options interlock: wife 1 in chair 4 forces 7→10, 3→6, 9→2, 5→8, and chair 8 forces the mirror. Only 2 fillings survive.

{1 → 4,7 → 10,3 → 6,9 → 2,5 → 8} or {1 → 8,5 → 2,9 → 6,3 → 10,7 → 4}
4STEP 4

Multiply the two stages

Each of the 240 men-seatings pairs with each of the 2 wife-fillings, so the total is 240 × 2 = 480 — choice (C).

240 × 2 = 480
Answer
480
Without the spouse rule, alternating seatings number 2 × 5! × 5! = 2 × 120 × 120 = 28800; the 'no neighbor, no across' rule slashes each block of 120 women-orderings down to just 2, giving 240 × 2 = 480. The result is a positive whole number, it is a multiple of the men-count 240 exactly as the ' × 2 for the women' structure predicts, and among the choices only 480 is such a multiple of 240 — all consistent with choice (C).
💡Key takeaway

Split a tangled seating count into stages — place one group first, then count how the constraints squeeze the second group — and multiply the stage counts together.

  • Seat the five men
  • Find each wife's legal chairs
  • Chain the wives: only two arrangements
  • Multiply the two stages

More like this

Same archetype — closest grade level first.