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Question Number 49982 by pieroo last updated on 12/Dec/18

$$\mathrm{If}\frac{{\mathrm{n}}_{{\mathrm{P}}_{\mathrm{r}}}}{{\mathrm{n}}_{{\mathrm{C}}_{\mathrm{r}}}}-1=5,\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{r}$$

Answered by ajfour last updated on 13/Dec/18

$${}^n{P}_r=\frac{n!}{(n-r)!}\mathrm{\&}{}^n{C}_r=\frac{n!}{r!(n-r)!}$$$$\Rightarrow \frac{{}^n{P}_r}{{}^n{C}_r}-1=r!-1=5$$$$\Rightarrow r!=6\Rightarrow r=3.$$

Commented by ajfour last updated on 13/Dec/18

$$2!=2,4!=24$$$$\Rightarrow 2!<r!<4!$$$$letscheck3!=1\times 2\times 3=6$$$$\Rightarrow r!=6\Rightarrow r=3.$$

Commented by pieroo last updated on 13/Dec/18

$$\mathrm{thanks}\mathrm{sir},\mathrm{But}\mathrm{how}\mathrm{do}\mathrm{i}\mathrm{explain}\mathrm{to}$$$$\mathrm{my}\mathrm{younger}\mathrm{brother}\mathrm{that}\mathrm{r}!=6\Rightarrow \mathrm{r}=3?$$

Commented by pieroo last updated on 13/Dec/18

$$\mathrm{wow},\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{for}\mathrm{your}\mathrm{patience}.$$

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