For a row of stations 1, 2, ..., 14, think of them as dots. Connecting any two dots forms a unique line. There are total of 14x13/2=91 such lines. In order to allow for any connections, at the peak of demand the bus needs to have 49 seats. This is because, suppose the dots are divided into two groups: 1~7, 8~14, then the number of lines connecting the two groups is 7x7=49. Other divisions requires fewer lines. For example, division of 1~6, 7~14 results in 6x8=48 lines.
But the bus only has 25 seats, not 49, so some of the 91 connections are not possible. Thus more refined study is called for. Still divide it to two groups: 1~7, 8~14. Within each group, the total intra-group connections is 7x6/2=21, while the maximal seat demand is merely 3x4=12. The bus, with 25 seats at its disposal, handles these connections perfectly.
The limitation comes for inter-group connections. At station 7, all intra-group connections are finished, and all intra-group passengers should get off bus by now. So all 25 seats can be used for inter-group travelling. That means we can have a maximum of 25 intra-group connections.
21 (intra-group) + 25 (inter-group) + 21 (intra-group) = 67
