There are 20 towns grouped into four zones with five 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?Choices:- 220 240 250 270

To solve this problem, we need to consider two scenarios: towns within the same zone and towns in different zones.

1. Within the same zone: There are 4 zones with 5 towns each. In each zone, every town is connected to every other town with 3 direct lines. The number of connections between towns in a zone can be calculated using the combination formula nC2, where n is the number of towns. So, in each zone, there are 5C2 = 10 connections. Since each connection has 3 direct lines, there are 103 = 30 direct lines in each zone. Since there are 4 zones, the total number of direct lines within the same zones is 304 = 120.

2. Between different zones: Each town is connected to every town in the other zones with 1 direct line. There are 20 towns, so each town is connected to 20-5 = 15 towns in other zones. So, for each town, there are 15 direct lines. Since there are 20 towns, the total number of direct lines between different zones is 15*20 = 300. However, this counts each line twice (once for each town it connects), so we need to divide by 2 to avoid double-counting. So, the total number of direct lines between different zones is 300/2 = 150.

Adding these two totals together, we find that the total number of direct telephone lines required is 120 + 150 = 270. So, the answer is 270.