Do you want the number of non negative integral solutions to, say,
If so, we can do a change of variables.
This gives us
The number of integral solutions without restriction would be
I'm trying to achieve the number of ways you can add three numbers x, y and z to be equal to a given integer.
For example, let's say the given integer is 10.
so I need to know how many ways are there to add up three numbers x, y and z to be equal to 10?
x+y+z = 10. The values of x, y and z would have to be either a positive whole number, or only a decimal number with .5
Some of the ways are 1) x=0, y=0, z=10 2) x=0, y=5, z=5 3) x=4.5, y=4.5, z = 1. This list can go on, but will still be infinite given that u can only use either a positive whole number or a decimal number with only .5 such as 0.5, 5.5, 9.5 etc.
Do you think is it possible? If so, is there a general way to achieve this for any given integer rather than just 10?
To explain. That is the number of non-negative integral solutions to .
If we divide each of those solutions by two we get one of your solutions.
So for a general solution.
If N is a positive integer and we want to count the number of ways to add k non-negative integers or numbers with the decimal part .5 to get N, the idea is the same as above.