Expert: Paul Klarreich - 2/25/2007

Question
how many combinations will the be if i roll three dice?

Answer
Questioner:   amelia
Category:  Advanced Math
 
Subject:  maths
Question:  how many combinations will the be if i roll three dice?
..................................
Hi, Amelia,

In this context, you must be careful in using the word 'combinations'.  Assuming you have used it correctly, you are considering  1,1,2 to be the same as 1,2,1 and 2,1,1.

In that case, you want to consider only cases where the numbers are in increasing order -- i.e. 2,3,5, but not  3,2,5.

So you check the case where:

First die is 1:  The cases are
  Second die is 1: 6
  Second die is 2: 5
  Second die is 3: 4
  Second die is 4: 3
  Second die is 5: 2
  Second die is 6: 1
1 + 2 + 3 + 4 + 5 + 6 =  6(6 + 1)/2

First die is 2:  The cases are:
  Second die is 2: 5
  Second die is 3: 4
  Second die is 4: 3
  Second die is 5: 2
  Second die is 6: 1
Total =  5(5 + 1)/2

First die is 3:  The cases are:
Total =  4(4 + 1)/2

First die is 4:  The cases are:
Total =  3(3 + 1)/2

First die is 5:  The cases are:
Total =  2(2 + 1)/2

First die is 6:  The cases are:
Total =  1(1 + 1)/2

[Note that that last one is just 1 case - 6,6,6.

Now the total is:

   6     k(k+1)
Summation ------ =
 k = 1     2

   6     k^2+k
Summation ------ =
 k = 1     2
1    6(6+1)(2(6)+1)   6(6+1)
--- [ -------------- + ------ ]
2          6            2

1   
--- [ (7)(13) + 3(7) ]
2        


1   
--- [ (7)(16) ]
2        

  
(7)(8)

= 56

which you can easily check.  Is this what you had in mind?  Or did you just want to count them?

Case  1 is 1,1,1.
Case  2 is 1,1,2.
Case  3 is 1,1,3.
Case  4 is 1,1,4.
Case  5 is 1,1,5.
Case  6 is 1,1,6.
Case  7 is 1,2,2.
Case  8 is 1,2,3.
Case  9 is 1,2,4.
Case 10 is 1,2,5.
Case 11 is 1,2,6.
Case 12 is 1,3,3.
Case 13 is 1,3,4.
Case 14 is 1,3,5.
Case 15 is 1,3,6.
Case 16 is 1,4,4.
Case 17 is 1,4,5.
Case 18 is 1,4,6.
Case 19 is 1,5,5.
Case 20 is 1,5,6.
Case 21 is 1,6,6.
Case 22 is 2,2,2.
Case 23 is 2,2,3.
Case 24 is 2,2,4.
Case 25 is 2,2,5.
Case 26 is 2,2,6.
Case 27 is 2,3,3.
Case 28 is 2,3,4.
Case 29 is 2,3,5.
Case 30 is 2,3,6.
Case 31 is 2,4,4.
Case 32 is 2,4,5.
Case 33 is 2,4,6.
Case 34 is 2,5,5.
Case 35 is 2,5,6.
Case 36 is 2,6,6.
Case 37 is 3,3,3.
Case 38 is 3,3,4.
Case 39 is 3,3,5.
Case 40 is 3,3,6.
Case 41 is 3,4,4.
Case 42 is 3,4,5.
Case 43 is 3,4,6.
Case 44 is 3,5,5.
Case 45 is 3,5,6.
Case 46 is 3,6,6.
Case 47 is 4,4,4.
Case 48 is 4,4,5.
Case 49 is 4,4,6.
Case 50 is 4,5,5.
Case 51 is 4,5,6.
Case 52 is 4,6,6.
Case 53 is 5,5,5.
Case 54 is 5,5,6.
Case 55 is 5,6,6.
Case 56 is 6,6,6.
There are 56 of these, as you can easily see.

[By this time, you're probably sorry you asked.]