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Question:

The employee is distributing 7 objects among 4 containrs (Xi's).

Assuming the containers are X1 + X2 + X3 + X4 = 7 where Xi >= 0 $\forall$ 1<= i <= 4.

Determine all integer solutions.

Logical Questions:

  1. How do we understand this question involves combination with repetition ?
  • My answer: Order does not matter, and we are allowed to place as many objects we can inside a container which suggests repetition is allowed, so we use combination with repetition $n + r - 1 \choose r$. Am I right?
  1. What is n and what is r ? (I don't understand this).
  • My answer: To me at first, I thought the number of containers were the limiting factor, so r = 4, and n = 7. However, the objects are the limiting factors I believe ?
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    I just need confirmation to see if I am on the right track. – 2017-02-19
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    I believe we consider a number as our r when it is the limiting factor. Am I right ? – 2017-02-19
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    I think that the objects are a limiting factor because you can only move 3 of them, because of that condition that says that every container should have at least 1 object inside. – 2017-02-19
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    @JuanIgnacioCuiule, I do believe the objects are the limiting factor, so r = 7. How do you know every container X1 ... X4 gets atleast one object ? I mean, to me I think you can have an empty container because it did not explicitly mention that every container needs to be filled. – 2017-02-19
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    "Assuming the containers are X1 + X2 + X3 + X4 = 7 where Xi > 0 ∀ 1<= i <= 4." It says that the sum of all should be 7 and that for every container (Xi) the value should be greater than 0 – 2017-02-19
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    In the previous case you were right, but the actual question has greater or equal to, so a container can be left empty. – 2017-02-19
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    Why can't this just be stars and bars instead of "combinations with repetitions"? – 2017-02-19
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    @Kourosh look at this [video](https://youtu.be/ZcSSI6VY1kM?t=238) , I think that it can help you. – 2017-02-19
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    Yes indeed, thank you! – 2017-02-19

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Solution:

  1. Any counting problem that does not need order, and repetitions are allowed for its objects, then you can use combinations with repetition. Equation: $n+r-1\choose r$

  2. Your limiting factor is your r, so in this question the limiting factor is your number of objects, 7, and your total, n is your number of containers.

    $\therefore$ $r = 7$, $n = 4$

    $\therefore$ $4+7-1\choose 7$ = $10\choose 7$