How many solutions does this equation have in $\mathbb{Z}_{\ge 0}$?

Question

How many solutions are in $\mathbb{Z}_{\ge 0}$ of this equation: $x+y+z=6$. I was thinking to solve with method of circles and lines i don't know if it's a good idea.

Answer 1

Circles and lines (often referred to as "stars and bars", though maybe this is USA-centric?) is a good idea. A linear arrangement of $6$ circles and $2$ lines is equivalent to a solution of your equation; the $2$ lines divide the circles into $3$ groups, corresponding to the values of the three variables. How many ways are there to arrange $6$ identical circles and $2$ identical lines in a row?

As you said in the comments, there are $\frac{8!}{6!\cdot 2!}$ ways, as there woul be $8!$ ways to arrange $8$ objects, but we divide by $6!$ since the $6$ circles are identical, and divide by $2!$ since the $2$ lines are identical.