AMC 10 · 2016 · #22

Grade 7 arithmetic
Archetype Overlap / Complement Counting └ Sub-archetype Complement by Subtracting Forbidden Arrangements
combinations-basiccomplementary-countingdouble-counting complementary-counting ↑ Prerequisites: combinations-basic
📏 Medium solution 💡 3 insights
Problem
In a round-robin tournament every team plays every other team exactly once, with no ties. Each team finished with 10 wins and 10 losses. Count the sets of three teams {A, B, C} that form a cycle of beatings: A beat B, B beat C, and C beat A.

Pick an answer.

(A)
385
(B)
665
(C)
945
(D)
1140
(E)
1330

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

Counting cycles head-on is hard, but every group of three teams falls into exactly one of two shapes. Either the three results loop (a cycle), or one team in the trio beat the other two (call it the 'top' team). Tool #16 (Count the Complement) turns the hard question into an easy subtraction: cycles equal all trios minus the trios that have a top team. The non-cycle trios are easy to count because each has exactly one top team, so tool #2 (Make a Systematic List) tallies them by walking through each team as a possible top and choosing two of the teams it beat. Tool #1 (Draw a Diagram) with arrows makes the two shapes visible so the split is convincing, and tool #4 (Introduce a Variable) first pins down how many teams there even are.

1STEP 1

Find how many teams there are

Each team played 10+10=20 games, one against each opponent, so n-1=20 and there are 21 teams.

n-1 = 10 + 10 = 20 → n = 21
2STEP 2

Count every trio of teams

Choosing 3 of the 21 teams and ignoring results gives (21 · 20 · 19)/6 = 1330 unordered trios in all.

(21 · 20 · 19)/(3 · 2 · 1) = 7980/6 = 1330
3STEP 3

See the only two shapes a trio can make

Among a trio's three games, the arrows either loop into a cycle or point out from one top team that beat both others — never both.

trio = cycle or exactly one team beat the other two
4STEP 4

Count the non-cycle trios

Each non-cycle trio has one top team; that team beat 10, so pick 2 of 10: 21 · C(10,2) = 945 non-cycle trios.

21 · C(10, 2) = 21 · (10 · 9)/2 = 21 · 45 = 945
5STEP 5

Subtract to get the cycles

Cycles are all trios minus the non-cycle ones: 1330 - 945 = 385, choice (A).

1330 - 945 = 385 = (A)
Answer
385
Check the split adds back up: 385 cycles + 945 non-cycles = 1330, which matches the total number of trios from step 2, so nothing was lost or double-counted. The non-cycle count also passes a sanity check: there are 21 teams, each the top of 45 trios, and 21 · 45 = 945. Choice (E) 1330 is the total trio count and (C) 945 is the non-cycle count, both traps for stopping early; the cycle count 385 is smaller than both, as it should be.
💡Key takeaway

Every group of three teams either loops or has one team that beat the other two, so count the easy 'one team on top' groups and subtract them from all the groups to find the loops.

  • Find how many teams there are
  • Count every trio of teams
  • See the only two shapes a trio can make
  • Count the non-cycle trios
  • Subtract to get the cycles

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