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Find number of ways in which 16 apples can be distributed among four persons so that each of them gets at least one apple.

I worked out like this: Since every person must get 1 apple, we give one apple to each beforehand. That leaves us with 12 apples. Now there are C(12, 4) ways to give these 12 apples to 4 peoples(Am I correct). This makes for a solution of 495, but the answer is given as 455. Help!!!

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up vote 2 down vote accepted

You’re right to distribute $4$ of the apples first, leaving $12$ to be distributed arbitrarily, but after that you’ve gone astray. This is a stars-and-bars problem, and as you’ll see at the linked article, the correct answer is


The explanation of the formula in the article is reasonably good, but if you have questions after reading it, feel free to leave a comment.

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