The most probable score is the power of x with the largest coefficient in

(x + x² + x³ + x⁴)⁶

which is bound to be midway between the smallest power of 6 and the largest power of 24.

Once I have known the solution, I can prove its correctness. But how (through a sequence of logical arguments) do I reach this solution in the first place?

I tried working backwards as follows; but it doesn't take me anywhere(or even if it does, I fail to see it...) :( -
I see that each coefficient in the expansion represents the number of combinations of 6 digits that include 'm' no. of 1's,'n' no. of 2's and so on. The correspondence of this with the problem is fairly obvious which proves the correctness of the solution to me. I also see that the the term (x + x² + x³ + x⁴)⁶ can be read as "Either x or x² or x³ or x⁴" done 6 times. Furthermore, powers instead of multiples have been chosen as we need their addition and not product. Also, the expansion eliminates distinction between the various permutation of a given set 6 values and leaves only the number of possible of ways in which each combination can occur.

Regards,
Deep G.
: )