# How many ways to distribute? [closed]

In how many ways $n$ things can be distributed among $5$ persons such that no one receives more than 9 things?

$n\le 45$ and $n\ge 5$

• Welcome to Math.SE! Please read this post and the others there for information on writing a good question for this site. In particular, people will be more willing to help if you edit your question to include some motivation, and an explanation of your own attempts. – GNUSupporter 8964民主女神 地下教會 May 14 '18 at 15:26
• @GNU supporter I tried all methods I can't understand it anyway – David May 14 '18 at 15:26
• This is a common kind of homework problem. Could you please mention the source and motivation of the problem, and what methods you have seen in class for solving this kind of problem? – Carl Mummert May 14 '18 at 17:17

One approach is to construct a generating function, for one person you would get $1+x+\ldots+x^9$ and for $5$ people you get $$f(x) = \left(1+\ldots+x^9\right)^5$$ and you seek the coefficient of $x^n$ in $f(x)$. One approach is to expand it into Taylor series ...
• @David The generating function $f$ is constructed in a careful way to be $f(x)=a_nx^n$ where $a_n$ is exactly the number you seek for any $n$... – gt6989b May 14 '18 at 15:31