How to calculate the 15 geometrical isomers of an octahedral complex with all ligands different?
Assign "a" to one of the six equivalent positions.  
Note that there are now 5 choices (b,c,d,e,f) for the opposing position.  
You are left with four equivalent equatorial positions.  Once a ligand is assigned to one of these four, there are 3 ligands remaining to select for the opposing equatorial position.  
Finally, there are 2 ways to assign the last two positions. The two possibilities are mirror images of each other.    


So $5\times3\times2= 30$ 

30 isomers (15 pairs of enantiomers).