SOLUTION: If three people roll a dice once, how many ways can the sum of the three people’s rolls add up to exactly 9?

Algebra ->  Permutations -> SOLUTION: If three people roll a dice once, how many ways can the sum of the three people’s rolls add up to exactly 9?      Log On


   

Question 1160018: If three people roll a dice once, how many ways can the sum of the three people’s rolls add up to exactly 9?
Found 3 solutions by Alan3354, ikleyn, greenestamps:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
If three people roll a dice once, how many ways can the sum of the three people’s rolls add up to exactly 9?
==============
1,2,6
1,3,5
1,4,4
---
2,2,5
2,3,4
---
3,3,3
============
7 ways IF 1,2,6 is the same as 2,6,1 etc.

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.

In other form, the question is


    How many solutions does this equation have

        x%5B1%5D + x%5B2%5D + x%5B3%5D = 9

    in positive integer numbers x%5B1%5D, x%5B2%5D and x%5B3%5D ?


The solution can be done using the "Stars and Bars method", which gives in this case the answer  

         C%5B9-1%5D%5E%283-1%29 = C%5B8%5D%5E2 = %288%2A7%29%2F2 = 4*7 = 28.


ANSWER.  28 different solutions   //  (accounting for all possible permutations, cases and sub-cases . . . )

Solved and answered.

-----------------

For the "Stars and Bars method", see the lesson
    - Stars and bars method for Combinatorics problems
in this site,

or, alternatively, this Wikipedia article

https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29

https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29



Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The "stars and bars" method used by another tutor is a good tool for problems like this where the numbers are larger, and where there are no restrictions on how large the numbers can be.

However, direct use of the stars and bars method for this problem gives a wrong result, because the integers are restricted to the range 1 to 6 inclusive. So counting the number of ways by enumeration is necessary.

I would assume, unlike one of the other tutors, that, for example, 1+3+5 is a different result than 3+1+5....


  combinations    number of

   with a sum   permutations

     of 9

  --------------------------

    1+2+6         3! = 6

    1+3+5         3! = 6

    1+4+4      3!/2! = 3

    2+2+5      3!/2! = 3

    2+3+4         3! = 6

    3+3+3      3!/3! = 1

  ------------------------

            total:    25



ANSWER:  There are 25 ways for the sum of the numbers on the three people's dice to be 9.