TITLE: How many different ways can a number be expressed as a sum of any number of integers when order matters?
QUESTION [8 upvotes]: How many different ways can a number $n \in \mathbb{N} $ be expressed as a sum of any number of positive numbers when order matters?
My solution:
Since I know, that $n$ can be represented as a sum of $k$ positive integers ${{n-1}\choose{k-1}}$ different ways, and $n$ can be represented as a sum of at most $n$ positive integers (the case where they are all $1$), then we can present $n$ as a sum of of any number of integers $\sum\limits_{k=1}^{n} {{n-1}\choose{k-1}}$ different ways.

REPLY [2 votes]: Imagine that you are climbing an $n$-step staircase, and can take any size of steps you wish. Let $W(n)$ be the number of ways to do it. We have $W(1)=1$.
We now express $W(n+1)$ in terms of $W(n)$. Either our first step is $1$, in which case there are $W(n)$ ways to finish the climb. Or else the first step is $a\gt 1$. In that case, by making the first step $a-1$, we get a way of climbing an $n$-step staircase, and all ways of climbing such a staircase can be obtained in this way.
Thus $W(n+1)=W(n)+W(n)=2W(n)$. It follows that $W(n)=2^{n-1}$ for all $n$.