If coefficients of $r$ th and $(r + 1)$th term in the expansion of ${(3 + 2x)}^{74}$ are equal, then $r$ is equal to:

$T_{r+1}{=}^{74}{C}_r{3}^{74-r}(2x{)}^r$
$\Rightarrow$coeffcient of $(r + 1)$th term is ${ }^{74}{C}_r{3}^{74-r}{2}^r$ 
We are given   ${ }^{74}{C}_{r-1}{3}^{74-r+1}{2}^{r-1}{=}^{74}{C}_r{3}^{74-r}{2}^r$

$\begin{array}{l}\Rightarrow \frac{3}{2}=\frac{74!}{r!(74-r)!}\cdot \frac{(r-1)!(75-r)!}{74!}\\ =\frac{75-r}{r}\\ \Rightarrow 3r=150-2r\Rightarrow r=30\end{array}$