Combination with repeats I've been stuck for a while on this question and haven't found applicable resources.
I have 10 choices and can select 3 at a time. I am allowed to repeat choices (combination), but the challenge is that ABA and AAB are not unique.
10 choose 3 is the question.
I have been working on a smaller set to find a formula. 3 choose 3.
I came up with 27 results (if order matters) and 10 results if order doesn't matter. AAA, AAB, AAC, ABB, ACB, ACC, BBB, BBC, BCC, CCC
How do I go about solving these problems.
My closest hypothesis is choices^slots / slots! == 3^3/3!

This approach works because 3 is small.
Case 1: 3 distinct objects - ABC
There are ${ 10 \choose 3}$ ways here.
Case 2: 2 distinct objects, 1 repeated twice - AAB
There are $ 10 \times 9$ ways here.
Case 3: 1 distinct object, 1 repeated thrice - AAA
There are $10$ ways here.

This process becomes tedious as 3 gets large. You should look up Burnside's lemma for the general case.
