Online Course Discussion Forum

Combinatorics question

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Combinatorics question
by Jialin Wang - Monday, November 7, 2022, 9:51 PM

If you have 5 numbered balls and 5 numbered boxes, then there are 5^5=3125 ways of putting them in the boxes. However, if you first do stars and bars and then determine what the balls' numbers are, the answer is (5+5-1)C5*5!  = 15120, which is not 3125. Where did the overcounting happen?

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Re: Combinatorics question
by Dr. Kevin Wang - Tuesday, November 8, 2022, 9:45 AM

If a few balls are in the same box, their order does not matter.  That's why it is not simply $5!$ in each case, and that is also why stars and bars cannot apply when the balls are distinguishable.