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I want to climb down a flight of ten stairs, taking one or many steps at a time. In how many ways can I do this if I am able to take as many steps at a time as I like?

I don't know the answer; I am getting $3$ such as:

$1,2,3,4,..., 10$ (one at a time)

$2,4,6,..., 10$ (two at a time)

$5,10 $ (five at a time)

this is correct/incorrect?

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    If you have to choose the number of steps you will take at a time, and stick with that number the whole way down, then the only thing you have missed is the possibility of taking 10 at a time. But if you are allowed to (for example) take 3 steps, then 4, then 3, well, that's a whole different problem. So, which is it?2011-12-08
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    @Gerry Myerson: "taking one or many steps at a time" probably indicate a variable number of steps at each time ... also this version seems to be the interesting one.2011-12-08
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    If we have total freedom, there are $512$ ways.2011-12-08
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    See the WP page on the [number of compositions](http://en.wikipedia.org/wiki/Composition_(number_theory)#Number_of_compositions)2011-12-08
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    @MaX, what you say is true, but I find it hard to believe that under that interpretation OP was only able to find 3 ways to come down the stairs.2011-12-08

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