If I'm reading you correctly, it is a matter of combinatorics. I may have interpreted you incorrectly but I agree with Urchin.
Think of this way, you have 3 options for a shirt (first slot), 3 options for pants (second slot), and three options for a tie (third slot).
You have three independent choices and three things to choose from for each choice. So yes, it is 3^3.
But the order-independent version can be thought to be equivalent to the following:
You have a giant(infinite) vat of ping-pong balls numbered 1 to 3, and choose 3 ping pong balls from there. You subsequently forgot which order you chose them. How many distinguishable possibilities are there?
It is 10, like Urchin mentioned, but how do you do it more generally? With choosing r balls from n types of balls? That is an interesting question, imo.
The short answer is it is C(n+r-1,r), where C is the choose function.
I tried typing the "star" and "bar" explanation but I didn't like the formating.
So, here is a
link.
Nocapszy, if you like these types of questions you'll like Discrete Math (which is the mathematics behind computer science).
The following source has explanations of the basics.
Discrete Math Project - Permutations with Repetition
Also note, that these basics come in four flavors:
1)permutations without replacement
2)combinations without replacement
3)permutations with replacement
4)combinations with replacement
You'll find that transforming your counting problem into one of these is the most likely source of answers. That along with
generating functions will fulfill most people's combinatorics type questions.