Do you want the number of non negative integral solutions to, say,
?.
If so, we can do a change of variables.
Let
This gives us
The number of integral solutions without restriction would be
Hi there,
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?
Then the answer is easy:
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.
Example: so
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.