Count combinations differing by fixed number of elements
The trick to this count is to fix the elements that will be different, then calculate how many different choices we have for this set of elements. For each selected element we will have one less choice then we did originally, and for the elements which are to be the same we have one choice for that character. Let $V$ be the vector of the character set counts for the positions. In your example we would have $V=[3,2,4,2,3,3]$ and let $t$ be the number of differences. For each t-tuple $x$ (set of t positions) we have  $\prod_{i \in x} (V[i]-1)$ strings which differ in exactly those places. The final answer is the sum of this for every t-subset of the positions. Since you character set differs for each position you will need to go through all of the t-tuples to solve this. That is not overly bad since though since the number of t-tuples is polynomial instead of exponential as checking all combinations could have been.