The number of subsets of size $5$ in $\{1,\dotsc,8\}$ is $\left(\begin{matrix}8\\5\end{matrix}\right)=56$.
These can be divided into (a) subsets that do not contain $8$, and (b) those that do contain $8$.
The sets of type (a) are just the subsets of size $5$ in $\{1,\dotsc,7\}$, as shown on the left. The number of these is $\left(\begin{matrix}7\\5\end{matrix}\right)=21$.
The sets of type (b) are obtained by adding $8$ to a subset of size $4$ in $\{1,\dotsc,7\}$, as shown on the right.
The number of these is $\left(\begin{matrix}7\\4\end{matrix}\right)=35$. This shows that $\left(\begin{matrix}8\\5\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)+\left(\begin{matrix}7\\4\end{matrix}\right)$.
In general, for $n\geq m>0$ we have $$ \left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}n-1\\m\end{matrix}\right)+\left(\begin{matrix}n-1\\m-1\end{matrix}\right) $$
These can be divided into (a) subsets that do not contain $8$, and (b) those that do contain $8$.
The sets of type (a) are just the subsets of size $5$ in $\{1,\dotsc,7\}$, as shown on the left. The number of these is $\left(\begin{matrix}7\\5\end{matrix}\right)=21$.
The sets of type (b) are obtained by adding $8$ to a subset of size $4$ in $\{1,\dotsc,7\}$, as shown on the right.
The number of these is $\left(\begin{matrix}7\\4\end{matrix}\right)=35$. This shows that $\left(\begin{matrix}8\\5\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)+\left(\begin{matrix}7\\4\end{matrix}\right)$.
In general, for $n\geq m>0$ we have $$ \left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}n-1\\m\end{matrix}\right)+\left(\begin{matrix}n-1\\m-1\end{matrix}\right) $$
