This is the "interpolation method":

First, the four schemes except B and C are sorted: A (4,4)

Four programs can divide the whole program into five parts (or five intervals). B and C are discontinuous, so they are distributed in two different intervals: C (5,2). (It can be understood as "inserting" B and C into the five intervals of the other four programs to ensure that they are not connected)

After determining the positions of B and C, B can come first or C can come first: A (2,2)

The last two steps together are understood as A (5,2), and the possible number of "B and C are not together" is A (4,4) × A (5,2).

A (3,3) × A (4,2) is the possibility that "B and C are not together, and A ranks first". Here, A must be the first one. As long as the other five schemes are considered, the process is the same as above.