Whilst doing permutation and combination homework this subtle issue confounded me.
When you have different groups, say 5 circles and 5 triangles, if you want all the circle triangle combinations it's $5 \times 5$.
However if you say want all the ways to chose a group of 2 circles from the 5, you get $\frac{5\times 4}{2!}$. Simply multiplying them doesn't give you combinations anymore, it gives you permutations so you must divide it out.
I can recognize this is mathematically true, but I don't feel it intuitively. Why are there no repetitions for two different groups when you multiply them? When I multiply things in a line, I'm used to it producing permutations (e.g How many ways to position 8 boys in a line? $8 \times 7 \times ... \times 1$)
It comes down to whether the objects are indistinguishable or not. In your case, a combination of circle $C_1$ then circle $C_2$ and a combination of $C_2$ then $C_1$ would be indistinguishable, since with the order reversed they still make up the same group.
Therefore, aside from the $5 \cdot 4$, you must also divide this by $2$, since for any circle there are exactly $2$ arrangements of the same group.