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!