<s>[INST] If I roll three dice at the same time, how many ways the sides can sum up to $13$? If I rolled $3$ dice how many combinations are there that result in sum of dots appeared on those dice be $13$?
 [/INST] Let (x, y, z) be the numbers showing on the 3 dice.
We want x + y + z = 13.
Assuming the dice are distinguishable, the possibilities are:
(1, 6, 6)
(2, 5, 6), (2, 6, 5)
(3, 4, 6), (3, 5, 5), (3, 6, 4)
(4, 3, 6), (4, 4, 5), (4, 5, 4), (4, 6, 3)
(5, 2, 6), (5, 3, 5), (5, 4, 4), (5, 5, 3), (5, 6, 2)
(6, 1, 6), (6, 2, 5), (6, 3, 4), (6, 4, 3), (6, 5, 2), (6, 6, 1)
So, there are 21 different combinations.
</s>