Is there any other way that doesn't involve manually listing out every way?
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1See here. There are some others questions about the same topic. – Masacroso Mar 01 '16 at 14:46
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1Please make the body of your Question as self-contained as possible, not relying on the title alone to bear the burden of problem statement. – hardmath Mar 01 '16 at 15:19
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The MGF of a uniform distribution over $\{1,\dots,6\}$ is $$\varphi(t) = \frac{e^t-e^{7t}}{6(1-e^t)}$$ so the MGF of the sum $X$ of $n$ independent such r.v.'s $X_1,\dots,X_n$ will be $$\Phi(t) = \mathbb{E}[e^{tX}]= \mathbb{E}[e^{tX_1}]^n= \left(\frac{e^t-e^{7t}}{6(1-e^t)}\right)^n$$ is that enough for you, or do you want a closed-form (provided there even is one)? The above will allow you, via manipulations and very enjoyable differentiations, to find the probability of any outcome.
(For what it's worth, such a distribution can also be seen as a very special case of $k$-SIIRV, for $k=6$).
Clement C.
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