Problem 14 · 2005 AMC 8
Medium
Counting & Probability
round-robin
The Little Twelve Basketball League has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many games are scheduled?
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Answer: B — 96.
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Hint 1 of 2
Two kinds of games behave differently — same-division (played twice) and cross-division (played once). Count each kind on its own, then add.
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Hint 2 of 2
Count matchups, not games-per-team, to dodge double counting: a same-division pair is C(6,2), and each plays 2 games.
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Approach: split into same-division and cross-division matchups
- Same-division: each division has C(6,2) = 15 pairs, each pair plays twice ⇒ 15·2 = 30 games per division. Two divisions: 60.
- Cross-division: every team in one division plays every team in the other once — 6·6 = 36 games (no pair is counted twice here, since the two teams are in different divisions).
- Total: 60 + 36 = 96.
- The key safeguard: a 'game' is one unordered pair, so count pairs once, not each team's schedule — that's how you avoid the doubling trap (192, choice E, is exactly that error).
Another way — one team's games, then halve:
- Pick any team: it plays its 5 division-mates twice (10 games) and all 6 cross-division teams once (6 games) — 16 games.
- All 12 teams: 12·16 = 192, but every game was counted by both its teams, so divide by 2: 192 ÷ 2 = 96.
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