The perpendicular distance from origin to the plane Passing through the points  $\widehat{i}+2\widehat{j}+3\widehat{k},$$2\widehat{i}+3\widehat{j}-4\widehat{k},$  $3\widehat{i}-4\widehat{j}+5\widehat{k}$ is

$\begin{array}{l}\overline{a}\times \overline{b}=-17\widehat{i}+10\widehat{j}-\overline{k}\\ \overline{b}\times \overline{c}=-\overline{i}-22\widehat{j}-17\widehat{k}\\ \overline{c}\times \overline{a}=-22\widehat{j}-4\widehat{j}+10\widehat{k}\\ \left[\overline{a}\overline{b}\overline{c}\right]=-96\end{array}$

Perpendicular distance from origin to the plane 

$\frac{\left|\left[\overline{a}\overline{b}\overline{c}\right]\right|}{|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|}=\frac{96}{8\sqrt{30}}=\frac{12}{\sqrt{30}}$