How many ways to arrange 2 distinct things in 3 ways

Question

I'm wondering how one would more rigorously figure out how to arrange 2 distinct objects in 3 spots. I believe the answer to be 8, since if we take one item to be a and the other to be b, you can run through the possibilities:

$$aaa, bbb, aab, bba, bab, aba, abb, baa$$

so the answer is 8.

How would one figure this out without manually running through all the possibilities? Further, if we denoted the number of distinct objects as k and the number of spots m, how would one arrive at the general formula?

Thanks.

Answer 1

Each of $3$ spots has $2$ choices, so total $2\cdot 2 \cdot 2=8$ ways. In general $k^m.$