If the coefficients of r^th and (r + 1)^th terms in the expansion of $(3+7\mathrm{x}{)}^{29}$ are equal, then r equals

We have ${\mathrm{T}}_{\mathrm{r}+1}{=}^{29}{\mathrm{C}}_{\mathrm{r}}{3}^{29-\mathrm{r}}(7\mathrm{x}{)}^{\mathrm{r}}=\left({ }^{29}{\mathrm{C}}_{\mathrm{r}}\times 3^{29-\mathrm{r}}\times 7^{\mathrm{r}}\right){\mathrm{x}}^{\mathrm{r}}$

Coefficient of $(\mathrm{r}+1{)}^{\mathrm{th}}$ term is ${ }^{29}{\mathrm{C}}_{\mathrm{r}}\times 3^{29-\mathrm{r}}\times 7^{\mathrm{r}}$

and coefficient of r^th term is ${ }^{29}{\mathrm{C}}_{\mathrm{r}-1}\times 3^{30-\mathrm{r}}\times 7^{\mathrm{r}-1}$

From given condition,

$\begin{array}{l}{ }^{29}{\mathrm{C}}_{\mathrm{r}}\times 3^{29-\mathrm{r}}\times 7^{\mathrm{r}}{=}^{29}{\mathrm{C}}_{\mathrm{r}-1}\times 3^{30-\mathrm{r}}\times 7^{\mathrm{r}-1}\\ \Rightarrow \frac{{ }^{29}{\mathrm{C}}_{\mathrm{r}}}{{ }^{29}{\mathrm{C}}_{\mathrm{r}-1}}=\frac{3}{7} \left( \frac{{ }^n{\mathrm{C}}_{\mathrm{r}}}{{ }^n{\mathrm{C}}_{\mathrm{r}-1}}=\frac{n-r+1}{r}\right)\\ \Rightarrow \frac{30-\mathrm{r}}{\mathrm{r}}=\frac{3}{7}\text{ or }\mathrm{r}=21\end{array}$