Since we know the third letter is going to be "G" anyway, it might be easier if we just look at the ways in which we can arrange the seven other letters. In other words, how many ways can we arrange "ORANISM"? Now it becomes a simple factorial problem.
7 things to be arranged:
7! = 7*6*5*4*3*2*1 = 5040
There are exactly 5,040 ways in which the letters of ORGANISM can be arranged such that the third letter is G.
