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?

Options

A. 72
B. 90
C. 96
D. 144

Answer

To solve the problem of determining how many direct telephone lines are required to connect the towns, we can break it down step by step.

### Step 1: Understand the Zones and Towns
There are 12 towns grouped into 4 zones, with each zone containing 3 towns.

### Step 2: Calculate Lines Within the Same Zone
For towns within the same zone, each pair of towns is connected by 3 direct lines.

- The number of ways to choose 2 towns from 3 in a zone is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
- Here, \( n = 3 \) (the number of towns in a zone) and \( r = 2 \) (the number of towns we are choosing).

Calculating \( C(3, 2) \):
\[ C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]
So, there are 3 pairs of towns in each zone.

Since each pair of towns in the same zone is connected by 3 lines, the total number of lines for one zone is:
\[ \text{Lines in one zone} = 3 \text{ pairs} \times 3 \text{ lines/pair} = 9 \text{ lines} \]

### Step 3: Calculate Total Lines for All Zones
Since there are 4 zones, the total number of lines for all zones is:
\[ \text{Total lines in all zones} = 4 \text{ zones} \times 9 \text{ lines/zone} = 36 \text{ lines} \]

### Step 4: Calculate Lines Between Different Zones
For towns in different zones, each pair of towns is connected by 1 direct line.

- The number of ways to choose 2 towns from the total of 12 is:
\[ C(12, 2) = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]
This means there are 66 pairs of towns in total.

### Step 5: Calculate Lines Between Different Zones
Now, we need to subtract the pairs of towns that are in the same zone from the total pairs to find the pairs that are in different zones.

- Each zone has 3 towns, and we already calculated that there are 3 pairs of towns in each zone. Therefore, for 4 zones:
\[ \text{Total pairs in same zones} = 4 \text{ zones} \times 3 \text{ pairs/zone} = 12 \]

Thus, the number of pairs of towns in different zones is:
\[ \text{Pairs in different zones} = 66 - 12 = 54 \]
Since each of these pairs is connected by 1 line, the total number of lines for different zones is:
\[ \text{Total lines between different zones} = 54 \text{ lines} \]

### Step 6: Calculate Total Direct Lines
Now, we can find the total number of direct lines required by adding the lines within the same zones and the lines between different zones:
\[ \text{Total direct lines} = 36 \text{ (same zone)} + 54 \text{ (different zones)} = 90 \text{ lines} \]

### Final Answer
Thus, the total number of direct telephone lines required is **90**.

---