Counting positive integral solutions to an equation How many positive integral solutions to $y_1+y_2+y_3+y_4+y_5+y_6=13$ are there?
I started with $y_1=8$ and distributed the rest to $y_2$ to $y_6$, but I feel like I will miss some of them by doing so.

Consider $$y_1+y_2+y_3+y_4+y_5+y_6=13$$ where $1\leq y_i\leq8$ for each $i\in\{1,2,3,4,5,6\}$. Make the substitution $z_i=y_i-1$ for each $i$. This gives us $$z_1+z_2+z_3+z_4+z_5+z_6=7$$ where $0\leq z_i\leq 7$ for each $i$. The number of non-negative integral solutions to this equation corresponds to the number of positive integral solutions to the original equation. Thus there are $${7+6-1\choose 7}={12\choose 7}=792$$ positive integral solutions to $y_1+y_2+y_3+y_4+y_5+y_6=13$.
