Nice solution, however I also tried doing this problem and got a different answer.
I viewed all the cards as different, so the number of ways to distribute the cards was 9!/3!/3!/3! = 1680.
I think 9C3 is the number of ways to give out 3 cards to one person, not all 3.
SImilar to what you did, I then assumed everyone recieved a red card, 6 ways of doing this.
The number of ways to distribute the remaining 6 cards was 6!/2!/2!/2! = 90.
90*6/1680 = 32.14%
=^._.^=