# Counting distinct positive valued k-tuples that sum to n where each entry can be no greater than some value.

This is motivated by the desire to count the number of ways two dice can form the sums 2,3,4,...,12 respectively.

We can safely use the stars and bars method for 2,3,4,...,7 where the number of ways would be $\binom{n-1}{1}$ for each instance, but this no longer works for a sum like, say, 8 which can be realized in theory by the ordered pairs (1,7) and (7,1), but can't be realized in practice since our experiment limits us to sums made using the integers 1-6.

So, in general, what method would we use for problems like these, and for this particular dice problem, is there a canonical method that's typically used?

Edit: I do realize it would be a simple to do something like make a 6x6 table where the Cartesian product of the two rolls maps to a sum, and we could just count them, but I'm hoping for a snazzier method.