Question

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?

• A. 72
• B. 90
• C. 96
• D. 144

Hint:

Use combination formulas for counting distinct pairs of towns in zones, then calculate the internal and external connections.

Answer 1

The Correct Option is C

The total number of ways to select 2 towns out of 12 is given by the combination formula: \[ \binom{12}{2} = \frac{12 \times 11}{2} = 66 \] Each zone contains 3 towns, and the number of ways to connect two towns within a zone is: \[ \binom{3}{2} = 3 \] Thus, for the 4 zones, the total number of connections within the zones is: \[ 4 \times 3 = 12 \] Therefore, the number of connections between towns in different zones is: \[ 66 - 12 = 54 \] Since each connection between towns in different zones requires 1 direct line, the total number of direct telephone lines required is: \[ 54 \times 1 + 12 \times 3 = 96 \]