Count the rectangles of all sizes and of all positions that are formed using segments in a grid with (m) horizontal and (n) vertical lines.
Solution. A rectangle is uniquely described by two distinct vertical lines and two distinct horizontal lines. Therefore we can just select these two lines and multiply these quantities by rule of product:
\begin{align*} \binom{n}{2}\binom{m}{2} \end{align*}
How many ways are there to rearrange the letters in the word "anagram"?
Solution. We can choose an arrangement of the letters in "anagram" in two steps. We first choose 3 of the 7 positions to be a's, then permute "ngrm" in the remaining positions. Thus, we have \begin{align*} \binom{7}{3} 4! \end{align*}
ways to choose an arrangement.