Hi Textot,
Thanks for your contribution.
I really liked the way you started your answer.
You are of course correct,
a+b+c = 22
where a,b, and c are ordered integers all greater or equal to 0
will have exactly the same number of solutions.
Your use of stars and bars was also interesting but unfortunately, it was not appropriate for this problem as it does not take into account that the triplet must be ordered.
This means that your answer is somewhere in the order of 3! or 6 times too big.
I used your simplification of a+b+c=22
and made a probability contour map of it.
Instead of a,b, and c
I used
x, y-x and 22-y (if you add these up you get 22)
Now these restraints come into play
\(x\ge0\qquad (1)\\~\\ y-x>x\\ y\ge2x\qquad(2)\\~\\ 22-y\ge y-x\\ -2y\ge -x-22\\ 2y\le x+22\\ y\le \frac{x}{2}+11\qquad (3)\\~\\ 22-y<22\\ y\ge0\qquad (4) \)
I graphed all these here https://www.geogebra.org/classic/nzxqxyyv
this is the pic.
Every point in this region will give a triplet solution.
LaTex:
x\ge0\qquad (1)\\~\\
y-x>x\\
y\ge2x\qquad(2)\\~\\
22-y\ge y-x\\
-2y\ge -x-22\\
2y\le x+22\\
y\le \frac{x}{2}+11\qquad (3)\\~\\
22-y<22\\
y\ge0