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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

$$\binom{12+4-1}{4-1}=\binom{15}3=455\;.$$

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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