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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
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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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