What if we phrase it this way...
We have a multiple choice questionnaire. There are five questions Which box does ball X go into? and for each question you can select one answer box A, box B, or box C.
box A box B box C
Which box does ball 1 go into? x
Which box does ball 2 go into? x
Which box does ball 3 go into? x
Which box does ball 4 go into? x
Which box does ball 5 go into? x
There's three possible answers to the first question. Regardless of the answer the answer to the first question, there's three possible answers to the second question. And so on. So we count $3^5$ possibilities.
It would involve factorials (or a falling factorial) if we instead were not allowed to use a box more than once. (It wouldn't work in this case: we have more balls than boxes.)
For example, suppose we changed the problem to: In how many ways can 3 different balls be distributed to 5 different boxes, when each box can hold at most one ball?
Then there will be $5$ ways to place the first ball, $4$ to place the second ball, $3$ to place the third, giving $5 \times 4 \times 3$ ways in total.