Suppose you have good friends seated around the dinner table who wish to toast one another's good health by clinking wineglasses. How many "clinks" will you hear? Each of the diners clinks glasses with his or her companions. You might at first think there would be clinks. But if you listen carefully—turn your computer's sound on—you will realize that this counts each clink twice, once for each clinkee. Dividing by 2 gives the correct number of clinks: .
Reference: S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 pp. 47–48.