Permutation/Combination Problem

I am thinking about the possible combinations of the results from 3 methods plus 1 lexicon.
So it will be 4 pronunciations per name.
How many possible combinations of this 4 pronunciation will be?
First, I think it is a piece of cake problem.
Then, I realize that it's not easy to write the formula for this solution,
even I can list all posibilities manually.

To make sure that I listed all combinations correctly,
I also listed 4 to the power of 4 (= 4 x 4 x 4 x 4) = 256 combinations,
then grouped them into categories.
Finally, it came up with 15 patterns as following:

1. {abcd} - all methods have all different pronunciations
2. {a},{bcd} - only TLII different
3. {b},{acd} - only CART different
4. {c},{abd} - only PbA different
5. {d},{abc} - only CMUDICT different
6. {ab},{cd} - TLII same as CART, PbA same as CMUDICT
7. {ac},{bd} - TLII same as PbA, CART same as CMUDICT
8. {ad},{bc}
9. {ab},{c},{d}
10. {ac}, {b},{d}
11. {ad}, {b},{c}
12. {bc},{a},{d}
13. {bd},{a},{c}
14. {cd},{a},{b}
15. {a},{b},{c},{d}

Note:

• a, b, c, d means the results of methods: Table lookupII, CART, PbA and CMUDICT.
• if the results are in the same set {} means the results have the same pronunciation.