Number of solutions to an equation

[praharmitra]

Given the following integer equation $$x_1 + x_2 + ... x_n = m$$ where $$x_i \geq 0$$ and $$x_i$$ is an integer for all i.

The number of solutions to the above equation is $$^{n+m-1}C_m$$

I was wondering if we could view this as a selection of $$m$$ objects from a selection of $$n + m - 1$$ objects.

Is there a 1-to-1 correspondence between a particular solution of the equation, and a particular selection of m objects from a selection of some n + m - 1 objects.

I hope I have made myself clear. I have tried to figure out such a correspondence, but in vain.

 

[Hurkyl]

How about a selection of n-1 objects?

 

[awkward]

This may help:

http://en.wikipedia.org/wiki/Stars_and_bars_(probability)

 

[praharmitra]

Hey, sorry I haven't replied back on my own thread. Was away.

Anyway, I figured out the required one-to-one correspondence myself. Its pretty much related to the link that @awkward has provided.

So, if anyone still has trouble understanding, I would be glad to explain.

Thanks a lot guys for your help.

 

[blue_raver22]

I can confirm that the number of solutions to an equation can be determined using combinatorics. In this case, the equation x_1 + x_2 + ... + x_n = m can be seen as a combination problem, where we are selecting m objects from a total of n + m - 1 objects. This is because each variable x_i represents an object, and the sum of all the variables must equal m. So, we can use the formula ^{n+m-1}C_m to calculate the number of combinations or solutions to this equation.

To answer your question, there is indeed a 1-to-1 correspondence between a particular solution of the equation and a particular selection of m objects from a selection of n + m - 1 objects. This is because each solution of the equation represents a unique combination of objects that add up to m, and vice versa. Therefore, the number of solutions to the equation can be determined by the number of combinations of m objects from a total of n + m - 1 objects.

I hope this helps clarify the relationship between the equation and the combinatorial approach to determining the number of solutions. If you have further questions, please let me know.