Find the number of non-negative integer solutions of $x+y+z=14$ given that $0 \leq x,y,z \leq 9$

I know that similar questions have been answered before. Anyway, this can be solved using generating sequences or using star and bar argument along with PIE.

My question is a bit different. I feel like this can also be somehow manipulated and solved with the help of a dummy variable. Please help me with that, if that's possible.


1 Answer 1


See that when you make $x\ge 9$ others are always less than $9$. So count all possiblities than count possiblites when one of the numbers is greater than $9$ and multiply it by $3$ and subtract from total.

So let $N$ be the number of solutions to $x+y+z=14: x,y,z\ge 0$

let $n$ be the number of solutions to $x'+y+z=5:x', y,z\ge 0$

Answer: $N-3n$

  • $\begingroup$ Be careful. If $x > 9$, then $x - 10$ is a nonnegative integer. If we let $x' = x - 10$, we obtain $x' + y + z = \color{red}{4}$. $\endgroup$ Jun 15, 2018 at 9:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.