President and treasurer must be girls. So you need to choose 2 girls from 10,
which can be done in C(10,2) ways  = (10*9)/(1*2) = 45 ways.
For each of these combinations, there are 2 ways in which the 2 girls can 
be allotted the 2 posts (e.g. Girl1 can be the Pres and Girl2 the treasurer,
and vice versa).
Hence possible combinations = 45*2

Similarly, VP and secretary must be boys. So you need to choose 2 boys from 8,
which can be done in C(8,2) ways = (8*7)/(1*2) = 28 ways.
For each of these combinations, there are 2 ways in which the 2 boys can 
be allotted the 2 posts (e.g. Boy1 can be the VP and Boy2 the secretary,
and vice versa).
Hence possible combinations = 28*2

Finally, each set of 2 girls can be paired with each set of 2 boys to form the
final set of officers. So the number of possible combinations = 45*2*28*2 = 

 ways.

Hope you got it :)