Need help understanding book description for a combinatoris problem

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In summary, the problem asks how many ways can the number 4 be written as the sum of 5 non-negative integers. The provided solution shows how this can be achieved by selecting a certain number of times from each of the 5 "boxes" labeled 1-5. However, there is confusion about how this approach may result in repeats.
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fleazo
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*combinatorics*, sorry for the typo in the title. The problem is as follows: In how many ways can we write the number 4 as the sum of 5 non-negative integers?

I have taken a screen cap of the solution that my book provides. Here it is

http://imgur.com/8BhxXPq

So I understand the concept of how they're explaining this - for example - one possible selection is I chose "box 1" 2 times, "box 2" 1 time, "box 3" 0 times, "box 4" 2 times, and "box 5" 0 times, this corresponds to : 2 + 1 + 2 = 5.

The confusion for me is, it seems like this approach would have a lot of repeats. For example - the selection of "box 1" 0 times, "box 2" 0 times, "box 3" 2 times, "box 4" 1 time, and "box 5" 2 times, would also produce : 2 + 1 + 2 = 5.I realize I'm missing something in the logic, but I'm not sure what.
 
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oops... I realize now this is a textbook problem and not allowed here. I'm moving it to the textbook problem section. Sorry about that.
 

1. What is a combinatorial problem?

A combinatorial problem is a type of mathematical problem that involves counting or listing all possible combinations of a set of objects or elements. It can also involve finding the total number of ways to arrange or select items from a given set.

2. What is the purpose of understanding a book description for a combinatorial problem?

Understanding a book description for a combinatorial problem can help you gain a better understanding of the problem, its scope, and the techniques used to solve it. It can also provide a roadmap for learning and applying combinatorial methods in your own research or work.

3. How do I approach a combinatorial problem?

Combinatorial problems can be approached using various techniques, including enumeration, generating functions, and graph theory. It is important to carefully read and understand the problem description, break it down into smaller sub-problems, and use appropriate methods and tools to solve each sub-problem.

4. Are there any real-world applications of combinatorial problems?

Yes, combinatorial problems have numerous real-world applications in various fields, including computer science, biology, economics, and physics. For example, they can be used to optimize resource allocation, analyze DNA sequences, and design efficient communication networks.

5. Is there a specific background or knowledge required to understand combinatorial problems?

While a strong foundation in mathematics is helpful, there is no specific background or knowledge required to understand combinatorial problems. However, having a basic understanding of concepts such as permutations, combinations, and probability can be beneficial in approaching and solving these problems.

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