How many ways are there to distribute $30$ indistinguishable objects into $6$ distinguishable boxes if there has to be at least $2$ objects per box? I can get how to do this if it was at least 1 object, but I'm not sure how to approach this problem? If I was to solve this using stars and bars, would I have to put a bar in a gap with two objects on either side?

  • $\begingroup$ Welcome to MSE. You'll get a lot more help if you show that you have made a real effort to solve the problem yourself, even if you haven't made much progress. What are your thoughts? What have you tried? How far could you get? Where are you stuck? This question will likely be closed if you don't add more context. Please respond by editing the question body. Clarifications don't belong in the comments. $\endgroup$
    – saulspatz
    Apr 12, 2021 at 3:07
  • $\begingroup$ The numbers in the title and question do not match. $\endgroup$
    – RobPratt
    Apr 12, 2021 at 3:19

1 Answer 1


Hint: it equals to the number of ways of putting 18 objects into the six boxes without restrictions.

  • $\begingroup$ But why 18? I tried to approach this problem with stars and bars and assumed that there has to be at least 2 between each bar, giving 14 options to choose from. $\endgroup$
    – zasshu
    Apr 12, 2021 at 3:42
  • 1
    $\begingroup$ Because 12 balls are spoken for to satisfy the requirement. You don't have to guess where those ones go. $\endgroup$ Apr 12, 2021 at 3:50
  • 1
    $\begingroup$ Ohh, that makes sense. So if the problem asked for at least 3 objects per box, would I be solving for putting 12 objects into 6 boxes since each box must have 3 and the objects are indistinguishable? $\endgroup$
    – zasshu
    Apr 12, 2021 at 3:54
  • 1
    $\begingroup$ @ZaEric correct. $\endgroup$
    – Igor Rivin
    Apr 12, 2021 at 3:55

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