How is this a property of Pascal's triangle? For all non-negative integers $k$ and $n$,
$$
\dbinom{k}{k} + \dbinom{k+1}{k} + \dbinom{k+2}{k} + \ldots + \dbinom{n}{k} = \dbinom{n +1}{k+1}
$$
How is this a property of Pascal's triangle? I do not see at all how it relates. I'm looking for an explanation, as opposed to some calculation with algebraic manipulations.

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.
