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 $13$?

At the lower division math level we can do the following easily given the low number of combinations:
1)list the number of potential combinations
116
265
355
364
454
2) Now we find out the numbers of way that we can arrange the listed numbers in which it is: 
3
6
3
6
3 respectively
thus when we add the numbers we get 21
