Mathematics - Permutation & Combination Question with Solution | TestHub

The number of distinct throws when exactly

$r(1\le r\le 6)$ numbers appear will be

${}^6{\mathrm{C}}_{\mathrm{r}}\times$ (the number of ways of putting n identical things into r distinct boxes with no box empty) $={}^6{C}_r\times {}^{n-1}{C}_{r-1}$

The total number of distinct throws $={\sum }_{\mathrm{r}=1}^6{}^6{\mathrm{C}}_{\mathrm{r}}^{\mathrm{n}-1}{\mathrm{C}}_{\mathrm{r}-1}$ $={\sum }_{r=1}^6{}^6{C}_r{}^{n-1}{C}_{n-r}={}^{n+5}{C}_n={}^{n+5}{C}_5$