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I need to distribute 100 similar balls in 10 distinct boxes but the maximum balls in one box can be 20.

x1+x2+...+x10 = 100

xi ≤ 20 for every 1 ≤ i ≤ 10. how do I continue from here?

RobPratt
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Husam
  • 35
  • What exactly do you mean by similar ? .2. Do you know h{w to do it without capacity restrictions ? 3. Will you edit in what you have tried so far ?
    • – true blue anil May 23 '23 at 12:02
  • by similar I mean non distinguishable . I know how to do it without restrictions but now the whole question changes, even the way of solving it – Husam May 23 '23 at 12:06
  • What technique do you use without capacity restrictions ? The way of solving doesn't necessarily change with each technique. – true blue anil May 23 '23 at 12:09
  • it would be by the stars and bars formula 100-1 choose 9-1 if there was no restriction – Husam May 23 '23 at 12:19
  • If you use stars and bars you will have to augment it with inclusion-exclusion, which you may have learnt ? – true blue anil May 23 '23 at 12:22
  • which is exactly my problem here because this is the first time I have this kind of question – Husam May 23 '23 at 12:29
  • You should include any work you have done on the problem in your question (such as finding the number of solutions when there are no capacity restrictions) and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering. This MathJax tutorial explains how to typeset mathematics on this site. – N. F. Taussig May 23 '23 at 12:30
  • By the way the formula for stars and bars over positive integers would be $\dbinom{100-1}{10-1}$. Further, why don't you first look up stars and bars with inclusion-exclusion on this very site first ? – true blue anil May 23 '23 at 12:49
  • Probably easier to do this by recursion (generating functions might be another alternative) – Henry May 23 '23 at 13:09
  • Does link that has a variety of approaches, and an encapsulated formula that you can adjust according to your figures help you solve your problem ? Study it carefully. https://math.stackexchange.com/questions/4262545/what-is-the-probability-that-the-sum-of-six-rolls-of-a-fair-die-is-divisible-by/4262614#4262614 – true blue anil May 23 '23 at 21:54