Short Answer 5/216
Comment
The question really is, how many different types of throws with 4 dice will give 21? You can count them
Pattern 1
6663
Pattern 2
5556
Pattern 3 [ The hard one]
6654
Patterns 1 and 2 give 4 each.
6 6 6 3
6 6 3 6
6 3 6 6
3 6 6 6
5553 will do the same thing
Both can be found by (4/1) as a combination.
Now for 6654 That gives 12
6 6 5 4
6 6 4 5
6 5 4 6
6 5 6 4
6 4 5 6
6 4 6 5
5 6 4 6
5 6 6 4
5 4 6 6
4 5 6 6
4 6 5 6
4 6 6 5
That should total 12
The total number of ways of getting 21 with this pattern is 12
Total successes
12 + 4 + 4 = 30
What is the total number of ways you can throw 4 dice?
Total = 6 * 6 * 6 * 6 = 1296
What is the probability of success?
30 / 1296 = 5 / 216