Four (4) fair dice are rolled. what is the probability that the total is 21

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