You're asking, for example, for the sum of all the indicated cells of Pascal's triangle
$$ \begin{matrix} \cdot
\\ \cdot & \cdot
\\ \cdot & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \bullet &
\\ \cdot & \cdot & \cdot & \bullet & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \cdot & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \cdot & \cdot & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot
\end{matrix} $$
which is the same thing as the sum of the cells
$$ \begin{matrix} \cdot
\\ \cdot & \cdot
\\ \cdot & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \circ
\\ \cdot & \cdot & \cdot & \bullet & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \cdot & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \bullet & \cdot & \cdot & \cdot & \cdot
\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot
\end{matrix} $$
because I've added in zero. Now, what do you know about the sum of adjacent cells of Pascal's triangle?
Of course, this method is still a calculation with algebraic manipulations, just organized in picture form.