How many ways can we get a number by addition if each part of the addition has to be smaller or equal to a set value? For example, if we need to get 5 with the largest number we can use being 3, we can use:


*

*3 + 2

*3 + 1 + 1

*2 + 2 + 1

*2 + 1 + 1 + 1

*1 + 1 + 1 + 1 + 1


Is there any way to find out the solution with any two numbers without calculating every one of them?

If you wish to find the number of ways to express a positive integer $n$ as the sum of $m$ positive integers ($m\le n$) and each of the positive integers must be less than or equal to a certain value $\alpha$ then the number of such integers will be the number of positive integral solutions to these equations:
$x_1+x_2=n$ 
$x_1+x_2+x_3=n$ 
$x_1+x_2+x_3+\cdots+x_m=n$
such that $1\le x_i \le \alpha$
for the equation $x_1+x_2=n$ 
the number of positive integral solutions is the same as the coefficient of $x^n$ in the expansion $(x+x^2+x^3+\cdots+x^\alpha)^2$
for the equation $x_1+x_2+x_3=n$ 
the number of positive integral solutions is the same as the coefficient of $x^n$ in the expansion $(x+x^2+x^3+\cdots+x^\alpha)^3$
for the equation $x_1+x_2+x_3+\cdots+x_m=n$
the number of positive integral solutions is the same as the coefficient of $x^n$ in the expansion $(x+x^2+x^3+\cdots+x^\alpha)^m$
So the total number of the solutions of all of these equations is the desired number.
This method of finding the coefficient in the expansion works because the term $x^n$ is formed only if in the multiplication, the powers of some terms get added to give the specified $n$.
For example in your question, $5$ has to be expressed as the sum of positive integers not greater than $3$
$x_1+x_2=5$
Coefficient of $x^5$ in the expansion of $(x+x^2+x^3)^2$
$=(x+x^2+x^3)(x+x^2+x^3)$
$=x^2+2x^3+3x^4+2x^5+x^6$
Thus,$[x^5]=2$
Hence, 5 can be expressed in 2 ways as the sum of 2 positive integers lesser than or equal 3, which are
$(2,3)$ and $(3,2)$ ( yes, this method gives us the ordered pairs)
You can do the same for rest of the equations.
In general as well, if you allow integers from a certain interval (not only positive) say, $ \alpha \le x_i \le \beta$ then the number of ways to express $n$ as sum of integers (order is considered) is
($[x^n]$ is the coefficient of $x^n$)
$$\sum_{m=2}^n [x^n]  (x^ \alpha+x^{\alpha+1}+x^{\alpha+2}+\cdots+x^ \beta)^m$$ 
