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$

Please help. I failed trying all methods

  • $\begingroup$ 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. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 May 14 '18 at 15:26
  • $\begingroup$ @GNU supporter I tried all methods I can't understand it anyway $\endgroup$ – David May 14 '18 at 15:26
  • $\begingroup$ 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? $\endgroup$ – 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 ...

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  • $\begingroup$ Can you show the proof why this works in this particular problem? $\endgroup$ – David May 14 '18 at 15:29
  • $\begingroup$ I cannot visualize it, how it works!! 😓 $\endgroup$ – David May 14 '18 at 15:30
  • $\begingroup$ @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 $n$... $\endgroup$ – gt6989b May 14 '18 at 15:31
  • $\begingroup$ I still don't get the reason to sum variable x in ascending powers. Can you help me with any link which would help me to explain why such generating function is required? $\endgroup$ – David May 14 '18 at 15:34
  • $\begingroup$ Sir I accepted your answer though I didn't understand why such function is required $\endgroup$ – David May 14 '18 at 15:44

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