If I have an infinite amount of identical red, orange, yellow, and green balls, how many unique bins of size $12$ could be made from $12$ of them (order of balls does not matter). For instance, one possible bin would consist of $3$ red, $3$ orange, $3$ yellow, $3$ green, and another possible bin could have $10$ red, $1$ orange, $0$ yellow, and $1$ green.
Initially, I thought it would just be $4^{12}$ since each of $12$ balls placed in a bin has $4$ different coloring options, but I think this would only be the answer if ordering mattered (which it doesn't).
My intuition is telling me to find the total amount of combinations of ratios between colors that can be made, but I'm lost on how I would go about counting that, or if that is even what I should be focusing on..?
Would it be $$\frac{12!}{1!2!3!4!}\:?$$ This would be my only other guess.