SOLUTION: 4)Eight policemen are to be posted to guard three separate buildings. In how many ways may they all be posted if no building is to be guarded by less than two policemen?

Question 366816: 4)Eight policemen are to be posted to guard three separate buildings.
In how many ways may they all be posted if no building is to be guarded by less than two policemen?

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
The partitioning can only take place in two ways: either 2,2,4 or 3,3,2.
For the first case, there are C(8,4)*C(4,2)*C(2,2) = 420 ways of partitioning 8 men in two groups of two and 1 group of 4. After partitioning the 3 groups can be assigned in 3! = 6 ways. Therefore there are 420*6 = 2,520 ways.
Similarly there are C(8,3)*C(5,3)*C(2,2) = 560 ways of partitioning 8 men in two groups of 3 and 1 group of 2, and 6 ways of assigning them to the different buildings. Therefore there are 560*6 = 3,360 ways.
Finally since the two cases are mutually exclusive, there are a total of 2,520 + 3,360 = 5,880 ways.
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