Find the number of positive integers solutions of the equation $3x+2y=37$ Find the number of positive integers solutions of the equation $3x+2y=37$ where $x&gt;0,y&gt;0,\ \ x,y\in \mathbb{Z}$  .


By trial and error I found

$$\begin{array}{|c|c|}  \hline
 x &amp; y \\ \hline
11 &amp; 2 \\ \hline
9 &amp; 5  \\ \hline
7 &amp; 8  \\ \hline
5 &amp; 11  \\ \hline
3 &amp; 14  \\ \hline
1 &amp; 17  \\ \hline
\end{array}$$

Total  $\large 6$  pair of solutions.
But i would like to know if their is a specific method to only find the number of positive solutions and not necessarily the actual solutions.

I look for a short and simple way.
I have studied maths upto $12th$ grade.
In the world of combinatorics, you can use something commonly known as the "stars and bars" method. Generally put, the equation $$x_1+x_2+\ldots+ x_k = s$$ where $s,x_i$ are positive integers has $\binom{s-1}{k-1}$ many solutions. The quantity  $\binom{s-1}{k-1}$ is called a binomial coefficient. A general binomial coefficient $\binom{a}{b}$ where $a,b$ are nonnegative integers with $a \geq b$ is defined as $$\binom{a}{b} =\frac{a!}{b!(a-b)!}$$ which means $$ \binom{s-1}{k-1} = \frac{(s-1)!}{(k-1)!(s-k)!}$$ In your problem we have $s = 37$ and $k = 2$. So, we want to calculate $$\binom{37-1}{2-1} = \frac{36!}{1!\space 35!} = 36$$ However, this tells us that the equation $x+y = 37$ has $36$ solutions. Since your equation is $3x+2y = 37$, we'll want every third $x$ and every second $y$. This is tantamount to dividing $36$ by $3$ and $2$. We get $\frac{36}{3\cdot 2} = 6$, which matches your calculation. (Had the ratio not been an integer, you'd want to round down to the nearest integer). Depending on how large $k$ and $s$ are, you may find the stars and bars method to be preferable to calculating every solution by hand. It really boils down to a pretty easy factorial calculation. I'd say your problem could go either way.
Considering: 3x + 2y = 37
It will have: Floor(Floor(37/3)/2)=6 positive integer solutions
Consider: 2y = 37-3x 
The max number of possible solutions is &lt; Floor(37/3)=12
That is: {1, 2, 3, ..., 12} are the possible solutions
But: 37-3x (mod 2) = 0 (as 2 must divide 37-3x)

So generally: ax + 2y = c
Will have Floor(Floor(c/a)/2) positive integer solutions