Let the average height of the original $22$ toddlers be $`x\text{’}$ inches.
$\dfrac{x_1 + x_2 + \ldots + x_{20} + x_{21} + x_{22}}{22} = x \quad \longrightarrow (1)$
Let $x_{21},x_{22}$ leaves the group.
$\dfrac{x_1 + x_2 + \ldots + x_{22}}{20} = x + 2$
$\Rightarrow x_1 + x_2 + \ldots + x_{20} = 20x + 40 \quad \longrightarrow (2)$
Also, $\dfrac{x_{21} + x_{22}}{2} = \dfrac{x}{3}$
$\Rightarrow x_{21} + x_{22} = \dfrac{2x}{3}\longrightarrow(3)$
From equation $(2)$ and$(3)$ , put the value in equation $(1)$, we get.
$\dfrac{20x + 40 + \frac{2x}{3}}{22} = x$
$\Rightarrow \dfrac{60x + 120 + 2x}{66} = x$
$\Rightarrow 62x + 120 = 66x$
$\Rightarrow 4x = 120$
$\Rightarrow \boxed{x = 30}$
$\therefore$ The average height of the remaining $20$ toddlers $ = x + 2= 30 + 2= 32$ inches.
Correct Answer$:\text{C}$