Suppose two of the eight locations are planned to be designated as core location. A core location is a location which is connected with every other location by direct bus each way, both in the morning as well as in the evening. The only direct bus which will be scheduled are originating in or terminating in one of the core location. Then, the minimum number of direct buses that need to be planned so that company is able to serve all the eight locations without visiting more than one core location during one trip are ... ?

Let's name the 8 locations as L1, L2, L3, L4, L5, L6, L7, L8.

Given there are two core locations, from each of the core location there has to be a bus to 7 other location.

Also, there has to be a bus from the other locations to the core location. So, for the first core location there are 7*2=14.

Similarly, for the second core location there has to be 14 buses. But this includes bus from first core location to second core location and bus form second core location to first core location. These two are already counted in the first. So, there will be 12 additional buses for the second location.

This is all for the morning. The same number of buses will be there for the evening. So, the total number of buses will be 2*(14+12)=2*26=52