SOLUTION: 8 students are standing in 2 rows of 4, facing the other row. If student A doesn't like facing or standing next to student B, how many different arrangements are there?

Suppose the positions in the 2 rows of 4 are numbered 1 through 8.

1 2 3 4
5 6 7 8

If there were no restrictions the answer would be 8!
However, A and B cannot be in any of these 10 positions:
1&2, 2&3, 3&4, 5&6, 6&7, 7&8, 1&5, 2&6, 3&7, or 4&8

So we must find the number of ways in which they are in these
positions so that we can subtract that number from the 8!
s.

We can choose the positions for A and B in 10 ways.
 
For each of those 10 ways they are 2 ways for A and B to be in
them. (For instance in the case of 1&2, A can be in 1 and B in 2,
or B can be in 1 and A can be in 2.)

For each of those ways of placing A and B there are 6! ways to arrange
the other 6 people.

So there are 10*2*6! ways A&B can be standing side by side or across from
each other.

Answer: 8!-10*2*6! = 25920 ways

Edwin