(a) No consecutive S's
Without S's, there are a total of 6 letters and 7 spaces.
My solution comes up as:
1. Choose 3 spaces from 7 spaces $$ {7 \choose 3} $$
2. Letters without S's and two O's $$ {6!/2!} $$
3. Then we have
$$ {7 \choose 3} ({6!/2!}) $$
(b) No consecutive vowels
1. Choose 4 spaces from 6 spaces $$ {6 \choose 4} $$
2. Organize 5 letters (without vowels) and 3 S's $$ {5!/3!} $$
3. Organize vowels, which include two O's $$ {4!/2!} $$
$$ {6 \choose 4} ({5!/3!}) ({4!/2!}) $$
(c) All S's come before any of the O's
1. Put S and O together sssoo and emplace them $$ {9 \choose 5} $$
2. Organize the rest of the letters $$ 4! $$
I ended up with this, but I'm not very sure whether I missed anything or not. Can you give me any hints if I missed anything?