If I were going to generate the set of sequences that add up to one -- without generating them all and testing -- I'd do it like this:
The positives and negatives in the sequence have to balance out. The sum of positive numbers in the sequence is one more than the absolute value of the sum of negative numbers. So if I start with a "correct" sequence, I can add one to a number, and subtract one from another, and the total remains 1.
1) Start with a sequence of 10 zeros
2) Pick one of those values and make it one (the first correct sequence)
3) Pick one answer that is less than 2 and add one. Then I'd pick one number that is greater than -2 and subtract one. (another correct sequence)
4) Repeat 3
If I did that methodically, I'd end up with the 837,100 correct sequences that Ross called out.
However, no one has ever accused me of being methodical, so what if I did it randomly?
1) Start with a sequence of zeros
2) Randomly pick one and make it one (a correct answer)
3) Randomly pick one that is less than 2 and add one. Randomly pick one that is greater than -2 and subtract one. (still a correct answer)
4) Repeat 3 about 9 times.
(...about 9 times because the maximum possible sum of positives is 10, with a sum of negatives of -9, but we added 1 up front. I played with it on my computer and repeated (3) up to 1000 times... the + and - balance each other out pretty quick, and each iteration is guaranteed to be correct anyway.)
Now, this doesn't satisfy the criterion that all the solutions must be equally likely ... I didn't test, but I don't see how they could be ... but I mentioned that criterion only because it seemed to simplify things.
But this seems more like a cheesy arithmetic trick than math. Ross's answer seems more correct mathematically.