Connecting Towns with Direct Telephone Lines

CAT 2003 Slot 1 · QA · Medium · Combinatorics

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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?

  1. A.

    72

  2. B.

    90

  3. C.

    96

  4. D.

    144

Answer

B

Explanation

In each zone of 3 towns, pairs of towns within the same zone = (32)=3\binom{3}{2} = 3. Each pair has 3 lines, so 3×3=93 \times 3 = 9 lines per zone. For 4 zones: 4×9=364 \times 9 = 36 lines. Pairs of towns in different zones = (122)−4×(32)=66−12=54\binom{12}{2} - 4 \times \binom{3}{2} = 66 - 12 = 54. Each such pair has 1 line = 54 lines. Total lines required = 36+54=9036 + 54 = 90.

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