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Question Number 134389 by EDWIN88 last updated on 03/Mar/21

$$\mathrm{Eight}\mathrm{dice}\mathrm{are}\mathrm{tossed}.\mathrm{If}\mathrm{the}\mathrm{dice}\mathrm{are}\mathrm{identical}\mathrm{in}$$$$\mathrm{appearance},\mathrm{how}\mathrm{many}\mathrm{different}-\mathrm{looking}$$$$\left(\mathrm{distinguishable}\right)\mathrm{occurrences}\mathrm{are}\mathrm{there}?$$

Answered by bramlexs22 last updated on 03/Mar/21

$$\mathrm{Theorem}$$$$\mathrm{If}\mathrm{r}\mathrm{is}\mathrm{nonnegative}\mathrm{integer},\mathrm{the}\mathrm{number}$$$$\mathrm{of}\mathrm{n}-\mathrm{tupples}({\mathrm{r}}_1,{\mathrm{r}}_2,...,{\mathrm{r}}_{\mathrm{n}}),\mathrm{is}{\mathrm{r}}_i\mathrm{an}$$$$\mathrm{integer}\mathrm{satisfying}\mathrm{the}\mathrm{condition}$$$${\mathrm{r}}_i⩾0(i=1,2,...,\mathrm{n})$$$${\mathrm{r}}_1+{\mathrm{r}}_2+{\mathrm{r}}_3+...+{\mathrm{r}}_{\mathrm{n}}=\mathrm{r}$$$$\mathrm{is}\left(\begin{array}{c}\mathrm{n}+\mathrm{r}-1\\ \mathrm{n}-1\end{array}\right)$$$$\mathrm{In}\mathrm{this}\mathrm{case}\to \{\begin{array}{l}\mathrm{n}=6\\ \mathrm{r}=8\end{array}$$$$\mathrm{so}\left(\begin{array}{c}6+8-1\\ 6-1\end{array}\right)=\left(\begin{array}{c}13\\ 5\end{array}\right)=\frac{13.12.11.10.9}{5.4.3.2.1}=1287$$$$$$

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