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Nicolas needs to distribute 14 boxes with his car in three tours(three times). It is supposed that in every tour he will distribute at least 1 box.

  1. On how many different ways can he do that, if all the boxes are the same?
  2. On how many different ways can he do that, if all the boxes are the same and his car can't carry more than 6 boxes?
N. F. Taussig
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Pol
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  • In the 1st problem, you want the number of non-negative integer solutions to $x_1 + x_2 + x_3 = 14.$ You have the added constraint that $x_1, x_2, x_3$ are each $\geq 1.$ The Stars and Bars theory in the first link covers both parts (1) and (2) of your problem. – user2661923 May 29 '22 at 17:37
  • There shouldn't be any constrains? Because it says that "It is supposed that in every tour he will distribute at least 1 box."? I am not sure I understand, I did read the two articles, but got to this solution that I wrote before. – Pol May 29 '22 at 17:44
  • Normal Stars and Bars theory is geared to each variable being a non-negative integer. In part (1), you have the added constraint that each variable must be $\geq 1$. Again, read the link. In part (2), you have the additional constraint that each variable must be $\leq 6$. Again, read the link, only this time, pay attention to the Inclusion-Exclusion portion of the answer, as well. I am specifically referring to the MathSE answer represented by the first link that I gave. – user2661923 May 29 '22 at 17:57
  • In the first problem, you want to find the number of solutions of the equation $x_1 + x_2 + x_3 = 14$ in the positive integers. See Theorem 1. In the second problem, you must exclude those distributions in which Nicolas takes more than $6$ boxes on one trip. That requires the use of the Inclusion-Exclusion Principle. If you solve the first problem, please add your solution to your question (not the comments). – N. F. Taussig May 29 '22 at 18:02