Textbook Problem

Think of a set with $m+n$elements as composed of two parts, one with m elements and the other with n elements. Give a combinatorial argument to show that $\left(\begin{array}{c}m+n\\ r\end{array}\right)\left(\begin{array}{c}m\\ 0\end{array}\right)\left(\begin{array}{c}n\\ r\end{array}\right)+\left(\begin{array}{c}m\\ 1\end{array}\right)\left(\begin{array}{c}m\\ r-1\end{array}\right)+\cdots +\left(\begin{array}{c}m\\ r\end{array}\right)\left(\begin{array}{c}n\\ 0\end{array}\right)$ where m and n are positive integers and r is an integer that is less than or equal to both m and n. This identity gives rise to many useful additional