In how many ways can a residence director assign six students to four dormitory rooms if two rooms are doubles and two rooms are singles ,and two of the students cannot be placed together?

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- September 20th 2005, 04:47 PMlycheeconfused with combination and permutation
In how many ways can a residence director assign six students to four dormitory rooms if two rooms are doubles and two rooms are singles ,and two of the students cannot be placed together?

- September 20th 2005, 10:33 PMrgep
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 6.5.4.3.2.1/2.2

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 2.4.3.2.1/2 are forbidden.

The answer is then 180-24 = 156.