Number of ways to divide a group of 20 mice 
A medical researcher needs to divide a group of 20 white mice into two control groups of 10 each. In how many ways can this be done?

I found the answer to be $\binom{20}{10}$.
What would have been the question if the answer were to be $\binom{20}{10}$ * 2?
 A: 
A medical researcher needs to divide a group of 20 white mice into two
  groups to conduct consecutive experiments. In how many ways can the
  researcher choose the group for the first experiment?

A: 
A medical researcher has $2$ initial groups of $20$ mice each. He needs to pick $1$ group and then divide it into $2$ control groups of 10 each. In how many ways can this be done?



*

*Step: $2$ ways to pick the initial group.

*Step: $\dbinom{20}{10}$ ways to pick $10$ mice from that product.


By the multiplication principle, the total number of ways to do it, is $$2\cdot\dbinom{20}{10}$$
A: We need to be crystal clear in the thinking
With both the mice and groups distinguishable (i.e. labelled)
There are $\binom{20}{10}$ ways to choose group $A$.
Group $B$ gets formed automatically, or we can write $\binom{20}{10}\binom{10}{10}$

With mice distinguishable, but groups indistinguishable
answer becomes $\binom{20}{10}/2$
Suppose e.g. there were two mice, $X$ and $Y,$ $\binom21$  represents choosing one mouse from the two, but it doesn't matter which we choose first, so there is only one way to divide here into two groups. 

Question to match answer of $\binom{20}{10}*2$
Both mice and groups are distinguishable,
and we want to select one of the groups for a particular experiment 
