# How to calculate the number of combinations that will add to a specific total

If I have for example 3 variables, and I need them to add to specific total, say 6, how do I calculate the number of combinations that will add to 6. Like 1+1+4=6 and 1+4+1=6 and 2+2+2=6 and 1+2+3=6 etc.

Is there a formula I can use? What if I am using say 10 variables with ranges from say 1 to 1000 and the total is for example 8063?

Thanks

You're trying to find the number of integer solutions to the problem: $$x_1+...+x_k=n, \ \ \ x_i\geq a$$ The general formula for number of solutions is $${n-ak+k-1 \choose k-1}$$. For three variable case, and positive integers as in your example, we have $$k=3,a=1,n=6$$, which yields $${5 \choose 2}=10$$ solutions. Specifically, they are $$\{(1,1,4),(1,4,1),(4,1,1),(2,2,2),\underbrace{(1,2,3)}_{6 \ \text{permutations}}\}$$.
The logic is simple. Let's say you want to solve $$x_1+x_2+x_3=6,\ \ x_i\geq 0$$. Each solution corresponds to a permutation of chracters ||111111, e.g. 11|111|1 corresponds to $$x_1=2,x_2=3,x_3=1$$. So, we'll use $$n$$ number of $$1$$'s and $$k-1$$ number of separators, which yields a total of $${n+k-1 \choose k-1}$$ permutations. A small extension is adding non-zero constraints, e.g. $$x_i\geq 1$$. Then, we first throw $$1$$'s into the variables, and allocate the remaining 1's. We can work out the formula for the general case, following the example.