There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required?

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Correct

140

150

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Official Explanation

Official

Consider a town A. It has 3 direct lines each with the other two towns in its zone (so 6 lines) and 9 direct lines with the 9 towns out of its zone. So A has 15 lines.

Since there are 12 towns, we have

15 \times 12

lines but here, each line is counted twice (since it connects 2 towns).…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.