1988 AIME Problems/Problem 1
One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below 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?
Currently there are possible combinations. With any integer from to , the number of ways to choose a set of buttons is . Now we can use the identity . So the number of additional combinations is just .
Note: A simpler way of thinking to get is thinking that each button has two choices, to be black or
to be white.
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