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Question Number 209281 by Tawa11 last updated on 06/Jul/24

$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{r},\mathrm{if}\stackrel{10}{}{\mathrm{C}}_{\mathrm{r}}=\stackrel{10}{}{\mathrm{C}}_{2\mathrm{r}+1}$$

Commented by klipto last updated on 06/Jul/24

$${}^{\mathbf{n}}{\mathbf{C}}_{\mathrm{r}}{=}^{\mathrm{n}}{\mathbf{C}}_{\mathrm{n}-\mathrm{r}}$$$${}^{10}{\mathbf{C}}_{\mathbf{r}}{=}^{10}{\mathbf{C}}_{9-2\mathbf{r}}$$$$\therefore \mathbf{r}=9-2\mathbf{r}$$$$\mathbf{r}+2\mathbf{r}=9,\mathbf{r}=3$$$$\mathbf{or}$$$$\mathbf{if}\mathbf{r}=2\mathbf{r}+1,\mathbf{r}=10-(2\mathbf{r}+1),\mathbf{r}=10-2\mathbf{r}-1$$$$\mathbf{r}+2\mathbf{r}=9,3\mathbf{r}=9,\mathbf{r}=3$$$$\mathbf{klipto}-\mathbf{quanta}$$

Commented by mr W last updated on 06/Jul/24

$$pleaseuse‘‘{answer}^{″}insteadof$$$$‘‘{comment}^{″}whenyouwanttoanswer$$$$aquestion.$$

Answered by A5T last updated on 06/Jul/24

$$\frac{10!}{r!(10-r)!}=\frac{10!}{(10-2r-1)!(2r+1)!}$$$$\Rightarrow (10-2r-1)!(2r+1)!=r!(10-r)!$$$$2r+1⩽10\Rightarrow r⩽4\Rightarrow r=3$$

Commented by Tawa11 last updated on 06/Jul/24

$$\mathrm{Thanks}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}.$$

Answered by A5T last updated on 06/Jul/24

$$r=2r+1orr=10-(2r+1)\Rightarrow r=3$$

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