Question

npr=npr+1 and ncr=ncr+1 find the values of n and r?

Answer 1

$$\begin{gathered} {\mathrm{P}}_{\mathrm{r}}{=}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{r}+1}\mathrm{and}{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}{=}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}+1}{ \\ }^{\mathrm{n}}{\mathrm{P}}_{\mathrm{r}}{=}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{r}+1} \\ \Rightarrow \frac{\mathrm{n}!}{(\mathrm{n}-\mathrm{r})!}=\frac{\mathrm{n}!}{(\mathrm{n}-\mathrm{r}-1)!} \\ \Rightarrow \frac{\mathrm{n}!}{(\mathrm{n}-\mathrm{r})(\mathrm{n}-\mathrm{r}-1)!}=\frac{\mathrm{n}!}{(\mathrm{n}-\mathrm{r}-1)!} \\ \Rightarrow (\mathrm{n}-\mathrm{r})=1(\mathrm{i})\mathrm{and} \end{gathered}$$

ⁿC_r =ⁿC_r+1

⇒ n!/ (r! (n - r)!) = n!/ ((r + 1)! (n - r - 1)!)

⇒ n!/ (r! (n - r) (n - r - 1)!) = n!/((r + 1) r! (n - r - 1)!)

⇒ (n - r) = (r + 1) --------(ii),


from (i) and (ii) we get r = 0, n = n!