Yeah probability questions are considered a bit on difficult side for GMAT.You can however solve these questions if you approach them logically.Every sentence in this question is giving an information which we can't ignore.

no country has more than one more runner than any other country
So fifteen runners can be distributed into 4,4,4 and 3 runners.If you consider any other distribution,you will violate above condition.
Now,these runners are participating in their national qualifying heat and only fastest of them are eligible to run in the international race.So,from four countries,these four runners then run a final race for first, second, and third place.
Cases possible:
Case 1.Runner from the nation having three runners initially come last and first, second, and third place went to runners who already ran in group of four.
Now,fastest runner from each nation can be selected in 4 ways each.
Total ways = 4*4*4
Now these runner's winning position can be arranged in 3! ways(viz. 123,132,213,231,312,321)
Total arrangement possible=3! 4*4*4 =384
Case 2 Runner from the nation having three runners initially didn't come last .So,one nation having four runners initially came last.
Now,we can select two nations of four runners from three nations of four in 3C2 ways=3 ways
Now,fastest runner from each nation can be selected in 4,4,3 ways each.
Total ways=4*4*3
Now these runner's winning position can be arranged in 3! ways
Total arrangements possible=3*4*4*3*3!
= 864
Adding,we get 384+864=1248