You can check your solution against these exact answers:
0 | _ | 738035/120932352 |
1 | _ | 586435/10077696 |
2 | _ | 4301435/20155392 |
3 | _ | 33057475/90699264 |
4 | _ | 33616325/120932352 |
5 | _ | 1145495/15116544 |
6 | _ | 737353/181398528 |
I don't think it's a particularly easy problem.
An exact formula for the probability of no matches is
6^-12 * Sum_{k=1..6}(6-k)^6 * C(6,k) * Sum_{j=0..k} (-1)^(k-j) * C(k,j) * j^n
where C(a,b)=a!/(b!*(a-b)!)
but this is a simpler case than the other numbers of matches.
Possibly proper mathematicians have more powerful tools to bring to bear on such problems.
Explaining even this simple formula would take quite a lot of work but if you're willing to crack open a maths book then "Stirling numbers of the second kind" would be a good place to start.
ByteCoin