Problem 3 · AMC 8 Stretch
Core
Counting & Probability
Geometry & Measurement
reduce-and-expandpattern-recognitionorganizing-datavisual-representation
There are 26 teams in the annual football draft. Each team's office has a direct phone line to every other team's office. How many phone lines are there in all?
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Answer: 325 telephone lines
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Hint 1 of 4
Drawing 26 offices at once is a mess. Start small. How many lines connect 1 office? 2 offices? 3? 4? Draw dots and connect every pair, then count the lines.
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Hint 2 of 4
Tabulate the line counts for 1, 2, 3, 4, 5 offices: you get 0, 1, 3, 6, 10. Now look at the JUMPS between them.
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Hint 3 of 4
The jumps are 1, 2, 3, 4, ... Each new office must connect to every office already there. So the \(n\)-th office adds \(n-1\) new lines, and the total is \(0 + 1 + 2 + \dots + (n-1)\).
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Approach: Reduce and expand — the handshake count \(\dfrac{n(n-1)}{2}\)
- Reduce the number of offices and count the connecting lines:
Offices Lines 1 0 2 1 3 3 4 6 5 10 - The jumps between line-counts are 1, 2, 3, 4, ... — every new office joins to every office already present, so the \(n\)-th office adds \(n-1\) lines, giving a total of \(0+1+2+\dots+(n-1) = \dfrac{n(n-1)}{2}\).
- Another view: each of the \(n\) offices needs a line to the other \(n-1\) offices, which is \(n(n-1)\) line-ends, but each line is counted twice, so divide by 2.
- For \(n = 26\): \(\dfrac{26 \times 25}{2} = \dfrac{650}{2} = 325\) telephone lines.
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