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 combinations and the principle of inclusion-exclusion to solve problems involving connections between groups.

Answer 1

The Correct Option is B

We have 12 towns, grouped into 4 zones, with 3 towns per zone. Step 1: For towns within the same zone, each pair of towns requires 3 direct lines. The number of ways to select 2 towns from 3 is: \[ \binom{3}{2} = 3 \] Since there are 4 zones, the total number of lines within the same zone is: \[ 4 \times 3 = 12 \] For each of the 12 lines within each zone, there are 3 direct lines, so the total number of direct lines within all zones is: \[ 12 \times 3 = 36 \] Step 2: For towns in different zones, each pair of towns requires only 1 direct line. The total number of pairs of towns from different zones is: \[ \binom{12}{2} - \binom{3}{2} \times 4 = 66 - 12 = 54 \] Thus, the total number of lines connecting towns in different zones is 54, with each line requiring 1 direct line. So, the total number of direct lines between different zones is: \[ 54 \times 1 = 54 \] Step 3: Total direct lines: \[ 36 + 54 = 90 \] Thus, the answer is: 90.