Those favoured are not necessarily to get into top 3. The top 3 may be all non-favoured, also can be all favoured, of course can be 2 favoured and 1 non-favoured, 1 favoured and 2 non-favoured.
The highest prob for a favourite pig to win also can be out of top 3 and at the same time other favourite pigs get into top 3.
So, is there the probability for favourite pig to win still needed?      

there are 455 combinations  
do we nid the permutation? if yes, 2730 combinations  

If this is like most toy problems, it shouldn't be a 15-variable problem. Assuming that pigs within the same class have same probability of winning, then there's only one variable involved - the bias in favor of the strong pigs vs the weak pigs.

The problem can be simplified by classifying the pigs into {Strong,Weak} x {Odd,Even}. And the top three positions as {Even,Odd} x {Even,Odd} x {Even, Odd} - pick only the outcomes that result in an even sum.

Then the rest of the problem can be solved with a probability tree. Okay, that would be how i'd do it coz i'm stupid - but it will work!