Balls and Ways

Let there be 1001 1001 balls in a bag.

The balls are numbered from 1 1 to 1001 1001 and you are allowed to choose 1 1 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?

2 500 + 1 2^{500}+1 2 1001 2^{1001} 2 500 2^{500} 2 500 1 2^{500}-1

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1 solution

Varsha Dani
Feb 20, 2019

There are 500 even numbered balls. Each of them may be in or out of your set, so there are 2 500 2^{500} 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 2 500 1 \boxed{2^{500}-1} .

Note that the fact that there are 1001 balls to begin with is a red herring.

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