# How many solutions are there to the equation $x_1 + x_2 + x_3 + x_4 + x_5 > 40$? [duplicate]

This question is an exact duplicate of:

The problem asks to find the number of solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 > 40$, where $x_i$ is an integer between $1$ and $13$ for $i = 1, 2, 3, 4, 5$ and such that no two "$x_i$"s are identical. ($x_i \neq x_j$ if $i\neq j$).

One can check that the sum of all the "$x_i$"s can not be greater than $55$. I suppose that one could solved the problem by summing up the amount of solutions for equations below : $$x_1 + x_2 + x_3 + x_4 + x_5 = 41$$ $$x_1 + x_2 + x_3 + x_4 + x_5 = 42$$ $$\cdots$$ $$x_1 + x_2 + x_3 + x_4 + x_5 = 54$$ $$x_1 + x_2 + x_3 + x_4 + x_5 = 55$$

But I do not know how to solve them.

Thank you in advance for your help!

## marked as duplicate by Did, Rohan, Roman83, Brevan Ellefsen, C. FalconDec 12 '16 at 0:42

This question was marked as an exact duplicate of an existing question.

• @Did a duplicate yes, but the other post got no decent answers. I vote to keep open for the time being. – Brevan Ellefsen Dec 11 '16 at 20:00
• @BrevanEllefsen Seeing that the same OP reposted this and that this is in explicit contradiction with the rules of the site, I would go as far as declaring that you are wrong in this matter. Sorry. – Did Dec 11 '16 at 20:30
• @Did ah, you're correct in that it's the same OP... I missed that fact. In that case I concede... You are correct, this ought to be closed. – Brevan Ellefsen Dec 11 '16 at 20:37