Calculate number of solutions in $N$ of diophantine inequality using generating functions.

Question

Calculate number of solutions in $N$ of diophantine inequality :

$x_{1}+x_{2}+x_{3}\leqslant 6$

such as:

$x_{1}>1$ $;x_{2}\leqslant2 ;2<x_{3}<5 $

Solve this using generating functions.

Can someone give me some tips for how to even start with this or give me solution to some example ? Thank you.

Answer 1

With the conditions imposed on $x_1$ and $x_2$, the only choice for $x_3$ is $x_3=3$

With $x_3=3$ , the only choices for $x_2$ and $x_1$ are$x_2=1$ and $x_1=2$

Thus the only solution is $$( x_1, x_2, x_3 ) = (2,1,3)$$