How many ways to place identical boxes in a row?

Question

How many ways there are to arrange ten (or n) identical boxes in a row of ten (or n) places when stacking is allowed? I guess you need to find how many different arrays there are but how?

Answer 1

Arrange the $n$ boxes in a row, like so: $$\square\ \square\ \square\ \square\ \square\ \square\ \square\ \square\ \square\ \square\ \square\ \square\ \ldots\ \square\ \square\ \square\ \square\ \tag{1}$$ There are $n-1$ spaces in between. "Stacking" the boxes means inserting a $|$ into some of the spaces in $(1)$, indicating when a new stack shall begin. Since there are $2^{n-1}$ ways to choose a subset of the $n-1$ spaces there are $2^{n-1}$ ways to arrange the boxes in a row of nonempty stacks.