Unique Dice Combinations : GMAT Quantitative Section
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# Unique Dice Combinations

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26 Dec 2013, 10:56
Hello all, I'm working on a dice game and need help. I have 6 die, all sides have unique words (36, 6 each die). I thought it was a simple factorial problem (6!=720) but I'm getting conflicting results. Does anyone have a way to generate all possible outcomes and the total number?
Thanks
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26 Dec 2013, 11:04
Six words to choose from die 1...
Six words to choose from die 2...
Six words to choose from die 3...
Six words to choose from die 4...
Six words to choose from die 5...
Six words to choose from die 6...

so 6*6*6*6*6*6 = 6^6 = 46656
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26 Dec 2013, 11:11
But there can only be 1 out come on each die per roll
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26 Dec 2013, 11:36
Fabman wrote:
But there can only be 1 out come on each die per roll

I'm not following.

Nevertheless, with combinatorial problems, the best thing to do is to first simplify the problem, figure out a formula, then scale up.

Let's say we have a three 3-sided dice (yes, such a thing doesn't exist, whatever!).
Die 1 has words A, B, C, die 2 has words D, E, F, and die 3 has words G, H, I.

Convince yourself (by writing down all combinations) that the total number of combinations is 3*3*3 = 27.
That will hopefully convince you that for your example, it will be 6^6.
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08 Jan 2014, 21:11
one die has 6 words (i'm gonna use numbers though)

1
2
3
4
5
6

now, if you throw 2 dice, you get the following combinations:

(assume you roll 1 on the first die)
1 and 1
1 and 2
1 and 3
1 and 4
1 and 5
1 and 6

so if you roll 1 on the first die, you get the 6 possible combinations above. if you instead roll 2 on the first die you get another 6 possibilities, and another 6 when you roll 3 on the first die, and again on 4, and on 5, and on 6. did you catch it? throwing an additional die takes all the previously existing combinations, and multiplies them by 6

so while it's obvious that 1 die gives 6 possible outcomes, 2 dice will give the same 6, but MULTIPLIED by 6 (so 6^2)
three dice will give that same 6^2 but MULTIPLIED by 6, for a total of 6^3
4 dice --> 6^4
5 dice --> 6^5
6 dice (your question) --> 6^6

hope this helps!
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Re: Unique Dice Combinations   [#permalink] 08 Jan 2014, 21:11
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