Well, without the restriction on the students who cannot be polaced together, there are 6.5/2 ways of placing students in the first double room, 4.3/2 of placing the remainder in the other double, 2 ways of placing the next student in the first single and then 1 way of placing the last student in the other single, giving 220.127.116.11.2.1/2.2
However, we need to count how many ways the forbidden pair go together. If they are in the first double, there are 4.3/2 times 2 times 1 ways of placing the rest: if in the second double, the same again. So 18.104.22.168.1/2 are forbidden.
The answer is then 180-24 = 156.