There's not a good reason to go with a complementary events strategy on this one. In fact, it will probably make the problem more complicated, though you did arrive at the correct answer.
I'm not, however, sure how you approached it specifically, so I cant comment on whether it was correct or whether you got lucky...
There are two ways he can get admitted in the first 2 tries.
Either he gets admitted the first time, or he gets rejected the first time and admitted the second time.
The odds of the first scenario:
The odds of the second scenario:
85% (rejection) * 20% = 17%
Add these together, since they're two different possibilities that fulfill our requirements, and we get 32%.