Explanation

(b)
In the expansion of ${(3 + 7x)}^{29}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T_{r + 1} = \ ^{29}C_{r} \bullet 3^{29 - r} \bullet (7x)^{r}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \left( \ ^{29}C_{r} \times 3^{29 - r} \times 7^{r} \right)x^{r}$
Let $a_{r}$= coefficient of $(r + 1)$th term
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ ^{29}C_{r} \times 3^{29 - r} \times 7^{r}$
$and\ a_{r - 1} =$coefficient of $r$th term
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ ^{29}C_{r - 1} \times 3^{30 - r} \times 7^{r - 1}$
According to question $a_{r} = a_{r - 1}$
$\Longrightarrow \ ^{29}C_{r} \times 3^{29 - r} \times 7^{r} = \ ^{29}C_{r - 1} \times 3^{30 - r} \times 7^{r - 1}$
$\Longrightarrow \frac{\ ^{29}C_{r}}{\ ^{29}C_{r - 1}} = \frac{3}{7} \Longrightarrow \frac{30 - r}{r} = \frac{3}{7}$
$\Longrightarrow 210 - 7r = 3r \Longrightarrow r = 21$