Problem 14 · 2012 AMC 8
Medium
Counting & Probability
handshake-counting
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
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Answer: B — 7 teams.
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Hint 1 of 2
Each team plays every other one, so a team plays N − 1 games. Multiply by the N teams — but careful, that counts each game from both sides, so you've double-counted.
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Hint 2 of 2
This is the handshake count: games (or handshakes) = N(N − 1)/2. The ÷2 fixes the double-count. Set it equal to 21 and find N.
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Approach: handshake count, then solve for N
- Each of the N teams faces the other N − 1 teams. Counting N × (N − 1) tallies every game twice (once for each team in it), so the real total is N(N − 1)/2.
- Set that to 21: N(N − 1)/2 = 21, so N(N − 1) = 42.
- Find two consecutive numbers multiplying to 42 — that's 7 × 6, so N = 7. (Faster than a formula: just scan products of neighbors.)
- Same pattern everywhere: handshakes in a room, diagonals plus sides of a polygon, pairs from a group — all are N(N − 1)/2 because every pair is counted from both ends.
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