Mathematics - Binomial Theorem Question with Solution | TestHub

${}^{39}{\mathrm{C}}_{3\mathrm{r}-1}-{}^{39}{\mathrm{C}}_{{\mathrm{r}}^2}={}^{39}{\mathrm{C}}_{{\mathrm{r}}^2-1}-{}^{39}{\mathrm{C}}_{3\mathrm{r}}$ $\Rightarrow {}^{39}{\mathrm{C}}_{3\mathrm{r}-1}+{}^{39}{\mathrm{C}}_{3\mathrm{r}}={}^{39}{\mathrm{C}}_{{\mathrm{r}}^2-1}+{}^{39}{\mathrm{C}}_{{\mathrm{r}}^2}$ $\Rightarrow {}^{40}{\mathrm{C}}_{3\mathrm{r}}={}^{40}{\mathrm{C}}_{{\mathrm{r}}^2}$ $\Rightarrow {\mathrm{r}}^2=3\mathrm{r}$ or ${\mathrm{r}}^2=40-3\mathrm{r}$ $\Rightarrow \mathrm{r}=0,3$ or -8,5 3 and 5 are the values as the given equation is not defined by $r=0$ and $r=-8$. Hence, the number of values of $\mathrm{r}$ is 2 .