How many ways can Amir, Bin, Charles, and Daria stand in a line if Amir and Bin refuse to stand next to each other?
Let's suppose that we can let anyone stand anywhere.....there are 4! = 24 ways to do this
If Ben and Amir DO stand next to each other they can occupy any 3 postions and be arranged in 2 ways for each of these = 3 * 2 = 6 ways
So....the number of ways they can be arranged where they ARE NOT standing next to each other =
24 - 6 = 18 ways