In how many ways can 4 (distinguishable) balls be put into 5 (distinguishable) boxes so that exactly 3 boxes are not empty?
Please provide detailed explanation.
Please note that you must provide options in PS questions.
4 distinct balls
5 distinct boxes - 2 must be empty
First choose the 2 boxes which will be empty in 5C2 ways.
Now you have 4 distinct balls and 3 distinct boxes. Since no box should be empty, the only distribution which will work is 2, 1, 1 i.e. 2 balls in one box and 1 each in other two boxes.
Choose the box which will have 2 balls in 3C1 ways. Choose the two balls that will go together in the box in 4C2 ways.
The other two balls will be distributed to the other two boxes in 2! ways.
Total ways = 5C2 * 3C1 * 4C2 * 2! = 360 ways
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