SOLUTION: Three dice are thrown simultaneously the probability of getting a sum six or less is

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Question 1064597: Three dice are thrown simultaneously the probability of getting a sum six or less is
Found 2 solutions by Fombitz, math_helper:
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!
There are or possible outcomes.
Look at the possible rolls that have 1 as the first roll,
(1,1,1)=3
(1,1,2)=4
(1,1,3)=5
(1,1,4)=6
(1,2,1)=4
(1,2,2)=5
(1,2,3)=6
(1,3,1)=5
(1,3,2)=6
(1,4,1)=6
When the first roll changes to 2, 4 of them are removed,
(2,1,1)=4
(2,1,2)=5
(2,1,3)=6
(2,2,1)=5
(2,2,2)=6
(2,3,1)=6
When the first roll changes to 3, 3 more are removed,
(3,1,1)=5
(3,1,2)=6
(3,2,1)=6
When the first roll changes to 4, 2 more are removed,
(4,1,1)=6
There are no rolls that begin with 5 or 6 in the solution set.
So out of 216 possible outcomes, 20 have a sum of 6 or less.
Answer by math_helper(2461)   (Show Source): You can put this solution on YOUR website!
Here is a table of the number of possible outcomes for each sum 3..18, condensed due to the symmetry of the possible outcomes: 
SUM                 n_ways                 SUM
 3                       1                  18
 4                       3                  17
 5                       6                  16 
 6                      10                  15
 7                      15                  14   
 8                      21                  13 
 9                      25                  12
10                      27                  11

The total possible outcomes is twice the sum of the middle column: 2*(1+3+6+10+15+21+25+27) = 216 (= 6x6x6)

To get 6 or less, we see that there are ways for that to happen.
Thus Pr{ sum is 6 or less on 3 dice thrown } = or approx

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