If the #2# girls to be separated by at least #3# boys, that means they can either be separated with #3# boys or with #4# boys. Let's look at each case individually.
Case #1#: Separated with #3# boys:
If we have, say, #6# seats numbered #1# through #6#, the girls can either be sitting on seats #1# and #5# or on seats #2# and #6#. These two cases are just reflections of another, so the total permutations for both will be the same. I'll calculate it once and then multiply by #2# to account for the two ways.
Imagine having girl A and girl B. They can sit in the order of AB or BA. Thus, #2# ways to order the girls.
Now for the boys. There are #4!=24# ways to order them. Thus:
#2*2*24=96# ways for Case #1#.
Case #2#: Separated with #4# boys.
It's the same idea, but there's only one way to seat the girls: seat #1# and seat #6#. Because of this, we don't have a #2# in the product.
There are still #2# ways to order the girls and #4!=24# ways for the boys.
#2*24=48# ways for Case #2#.
Adding up the total number of ways, we get #96+48=144# total ways, assuming each boy and each girl are unique.
