Dice sum of the points $12$

Asked Aug 11 '22 at 17:13

Active Aug 11 '22 at 17:23

Viewed 54 times

$3$ dice are rolled. In how many of the possible $6^3$ outcomes is the sum of the points $12$?

Attempt: I would like to solve this problem using stars and bars. I arrived at the equation $x+y+z=12$. And to ensure that none of them have a value of zero, I feared $a=x+1$, $b=y+1$, and $c=z+1$. So we will have $a+b+c=9$. I thought of the following reasoning: "How many 3-digit numbers can be formed so that the sum of their digits equals 9?"

But I don't understand how to solve it, or if even reasoning would help to solve this problem by this method. The problem's answer is $25$.

edited Aug 11 '22 at 17:23

asked Aug 11 '22 at 17:13

Lambert macuse

4,264

1

See this answer. – user2661923 Aug 11 '22 at 17:28
2

Well, you can certainly just enumerate the examples. If the first die is $1,$ how many ways can the other dice add up to $11?$ Etc. – Thomas Andrews Aug 11 '22 at 17:39
2

Stars and bars doesn't work well when there is an upper limit on the size of the numbers. Well, it can be made to work, using inclusion/exclusion and stars and bars. But inclusion/exclusion is more complicated. – Thomas Andrews Aug 11 '22 at 17:40
1

In particular, your three digit number question can be solved with stars and bars, but not if you restrict the digits or $0,1,2,3,4,5.$ – Thomas Andrews Aug 11 '22 at 17:42
2

The inclusion-exclusion gives $\binom{9+2}{2}-3\binom{9+2-6}{2}=25.$ – Thomas Andrews Aug 11 '22 at 17:44

Dice sum of the points $12$

0 Answers0