I am stating very clearly that I do not know how to do this problem. I am just thinking out loud.
I am also goint to simplify the question to 18 items instead of 30
How many ways can 18 different objects be paired.
This is probably too simplistic but I am thinking that I could chose one item randomly and pair it with any of 17 others.
Then I could chose another item randomly and pair it with any of 15 others. etc
so that would be
17*15*13*11*9*7*5*3*1 = 34 459 425 ways
Now say there are 6 red, 6 blue and 6 green but they are labelled blue1, blue2 etc so they are all distinct.
How many ways can I pair all the red together, all the blue together and all the green together?
5*3*1 * 5*3*1 * 5*3*1 = 3*5*3*1 = 45 ways
How many ways can I have the red altogether?
5*3*1*11*9*7*5*3*1 = 155 925
How many ways can I have the green altogether?
5*3*1*11*9*7*5*3*1 = 155 925
How many ways can I have the blue altogether?
5*3*1*11*9*7*5*3*1 = 155 925
So how many ways can I have 3 pairs of the same colour
3*155 925-2*45 = 467685
So the likelihood that there are at least 3 pairs the same colour is
467685/34 459 425 = 0.0135720488661665
Now the original question also wanted to included triples of RG.RB, and GB
This is getting much more complicated and I am not at all confident that what I have already presented is correct anyway.

Anyway, Chrisof does that give you any ideas. I am not saying that what I have presented is correct. It might be total rubbish.....
If you would like to discuss what you think of my simplified question and answer (good or bad) then please feel free to do so.