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