Originally Posted by
Soroban Hello, milkntea!
Here's a primitive approach . . .
Suppse $\displaystyle n = 7$ . . . We have 7 objects.
Place them in a row with a space between then: .$\displaystyle o\;\_\;o\;\_\;o\;\_\;o\;\_\;o\;\_\;o\;\_\;o$
Select two of the spaces and insert "dividers".
So that: .$\displaystyle o\;|\;o\;o\;o\;o\;|\;o\;o$ .represents $\displaystyle 1+4+2$
. . .And: .$\displaystyle o\;o\;o\;|\;o\;o\;|\;o\;o$ .represents $\displaystyle 3+2+2$
Hence, there are: .$\displaystyle {6\choose2} \,=\,15$ possible solutions.
I'll leave it to you to list them . . .
.