For example, say I have $3$ objects, $6$ slots, and each object must be chosen twice, how do I go about solving that?
Would it just be $\binom{6}{2} \binom{4}{2} \binom{2}{2}$ or am I thinking about it completely wrong?
Additionally, say we changed it so object $A$ shows up $3$ times, object $B$ $2$ times, and object $C$ $1$ time.
I don't really understand the intuition behind it, so an explanation of the thought process would be awesome.
The way you go about thinking about this problem is that you have 6 positions at the start. Then you need to choose 2 of those 6 for the first type, then 2 of 4 for 2nd, and then 2 of 2 for third. So the end result is $\binom{6}{2} \binom{4}{2} \binom{2}{2}$