SOLUTION: How many ways are there to distribute 15 district objects into 5 distinct boxes with: i) At least three empty box ii) No empty box

Question 1195029: How many ways are there to distribute 15 district objects into 5 distinct boxes with:
i) At least three empty box
ii) No empty box
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

i) (part 1): 5 empty boxes

Obviously that can't be done... 0 ways

i) (part 2): 4 empty boxes

Choose 1 of the 5 boxes to be the one that is not empty: 5 ways
For each of the objects, there is only 1 choice for which box to go into: 1^15 = 1 way
Number of ways: 5*1 = 5

i) (part 3): 3 empty boxes

Choose 2 of the 5 boxes to be the ones that are not empty: C(5,2) = 10 ways
Each of the 15 items can go in either of 2 boxes: 2^15 ways:
Number of ways: 10*2^15

i) ANSWER: 5 + 10*2^15

ii) Since no box can be empty, first put one object into each of the 5 boxes: C(15,1)*C(14,1)*C(13,1)*C(12,1)*C(11,1) = 15*14*13*12*11 ways
Each of the remaining 10 objects can be put in any of the 5 boxes: 5^10 ways

ii) ANSWER: (15*14*13*12*11) * 5^10

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