Problem:
One commercially available ten-button lock may be opened by depressing - in any order - the correct five buttons. The sample shown at right has as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?
Solution:
There are configurations of the ten buttons. We are to exclude those in which none or all of the buttons are depressed, as well as those in which exactly five buttons are depressed. Thus the total number of additional combinations is
The problems on this page are the property of the MAA's American Mathematics Competitions