Find # of Solutions for ΣXi <= C: Combinatorics/Integrals

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In summary, the conversation discusses finding the number of solutions for a summation of N variables that are positive integers and equal to or less than a constant number, using combinatorics or integrals. The polytopic numbers are mentioned as a solution to this problem, but the explanation for their connection is not clear.
  • #1
qbslug
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How do you find the number of solutions for the summation of N variables called Xi, which must be POSITIVE INTEGERS, that are equal or less than some constant number say C.

ΣXi <= C
i = 1,2...N
Xi = 0,1,2,...
I need the number of solutions for this equation

note that "i" is the dummy variable and the subscript for the variable X
do you have to use combinatorics or can you approximate with integrals?
 
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  • #2
Yeah, this would be a messy combinatorial problem. Lucky for you, someone's already done the hard work. What you want to know about is the polytopic numbers.

The answer is surprisingly simple, though I'm not sure of a good intuitive way to explain why it is. I'm sure that someone else can enlighten us with a neat combinatorial proof.
 
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  • #3
I don't see the connection between the number of solutions to the equation and these
polytopic numbers
 

1. How do you calculate the number of solutions for a combinatorics problem?

The number of solutions for a combinatorics problem can be calculated by using the combination formula, nCr = n! / (r!(n-r)!). This formula is used when order does not matter and there is no repetition in the selections.

2. What is the significance of ΣXi in the problem?

ΣXi represents the sum of all the individual elements in the set. In combinatorics, it is often used to represent the number of combinations or selections from a set.

3. How is the value of C determined in this problem?

The value of C is determined based on the constraints given in the problem. It represents the maximum value that the sum of the elements in the set can have in order for the solution to be valid.

4. Can this problem be solved using integrals?

Yes, this problem can be solved using integrals. The integral of a function can be used to calculate the area under the curve, which can be applied to combinatorics problems to find the number of solutions.

5. Are there any shortcuts or tricks for solving this type of problem?

There are some common strategies that can be used to solve combinatorics problems, such as using the combination formula, creating a visual representation (such as a tree diagram or a table), or breaking the problem into smaller, simpler parts. However, there is no one-size-fits-all shortcut for solving these types of problems, and it often requires careful analysis and logical thinking.

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