Whenever people are used in combinations problems, they are usually assumed to be non-identical, for obvious reasons. (People are unique.)
With that assumption, the simplest approach I can see is to consider all possible ways to choose just the girl(s), and multiply by the number of ways to choose the boys.
A straightforward way to count ways to choose the girls is just C(4,1) + C(4,2) + C(4,3) + C(4,4).
This is the same as 2^4 - C(4,0) = (2^4)-1. Can you see why?
Then we multiply this by the number of ways to choose boys, which is 2^6. Can you see why?
So the overall answer is ((2^4)-1)*(2^6).