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A room has $10$ doors. I want to find the distinct possibilities to open this room.

My approach. For each door we have two distinct possibilities. Thus we have $2^{10}-1$ possibilities, bacause all doors may be opened. Is this approach correct?

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Yes, it's exactly right. You have one way for every non-empty subset of the doors. – Brian M. Scott May 4 '12 at 19:18
@BrianM.Scott: Please, if you do not mind, add your comment as an answer. I will accept it. – spohreis May 4 '12 at 19:19
Done! $\qquad\,$ – Brian M. Scott May 4 '12 at 19:20
up vote 1 down vote accepted

Yes, it's exactly right. You have one way to open the room for each non-empty subset of the doors, and there are $2^{10}-1$ such subsets.

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