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Say you have two 6 sided dice numbered from 1 to 6. How would you calculate the probability of rolling 2, 3, 4, ..., 12?

My professor told me the generating function $(x+x^2+x^3+x^4+x^5+x^6)^2$ is incorrect for starting the problem because the dice are treated as distinct. That I should be looking for the answer using identical dice and "sum". How would this be solved?

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  • $\begingroup$ If the dice are treated as distinct, why does your title describe them as identical? $\endgroup$ – N. F. Taussig Dec 21 '18 at 23:23
  • $\begingroup$ @N.F.Taussig My professor asked for identical dice while I provided distinct dice. $\endgroup$ – Violet Jung Dec 22 '18 at 1:58
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    $\begingroup$ The coefficients of that generating function (when divided by $6^2$ for normalization), are the probabilities in question. $\endgroup$ – Math1000 Dec 22 '18 at 2:41
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Well if your dice are identical then you can simply say that for finding the probability of rolling a certain sum $s$ we need to find the number of ways $x_1 + x_2$ can be equal to $s$ or rather we need to find the number of integral solutions to the equation $$x_1 + x_2 = s$$ where $$x_1,x_2 \equiv 1,2,3,4,5,6$$ We can solve this rather easily and then we can use the argument for probability.

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