Let there be balls in a bag.
The balls are numbered from to and you are allowed to choose or more balls in such a way that all the balls you choose are of even parity .
In how many ways can you do this?
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There are 500 even numbered balls. Each of them may be in or out of your set, so there are 2 5 0 0 ways to choose a set of them. However one of these is the empty set, which is precluded by the requirement that you must choose 1 or more balls. Hence the answer is 2 5 0 0 − 1 .
Note that the fact that there are 1001 balls to begin with is a red herring.