If the vectors $\overrightarrow{\mathrm{a}}=\mathrm{\lambda }\widehat{\mathrm{i}}+\mathrm{\mu }\widehat{\mathrm{j}}+4\widehat{\mathrm{k}},\overrightarrow{\mathrm{b}}=-2\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}-2\widehat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{c}}=2\widehat{\mathrm{i}}+3\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$ are coplanar and the projection of $\overrightarrow{\mathrm{a}}$ on the vector $\overrightarrow{\mathrm{b}}$ is $\sqrt{54}$ units, then the sum of all possible values of $\mathrm{\lambda }+\mathrm{\mu }$ is equal to 

$\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}}$ are co planar
$\begin{array}{l}\Rightarrow \left|\begin{array}{ccc}\mathrm{\lambda }& \mathrm{\mu }& 4\\ -2& 4& -2\\ 2& 3& 1\end{array}\right|=0\\ \Rightarrow \mathrm{\lambda }\left(10\right)-\mathrm{\mu }\left(2\right)+4(-14)=0\\ \Rightarrow 5\mathrm{\lambda }-\mathrm{\mu }-28=0\dots \dots .\left(1\right)\end{array}$
Projection of $\overline{\mathrm{a}}\text{ and }\overline{\mathrm{b}}=\sqrt{54}$
$\begin{array}{l}\Rightarrow \frac{|\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}|}{\left|\overline{\mathrm{b}}\right|}=\sqrt{54}\\ \Rightarrow \frac{-2\mathrm{\lambda }+4\mathrm{\mu }-8}{\sqrt{24}}\pm \sqrt{54}\\ \Rightarrow -2\mathrm{\lambda }+4\mathrm{\mu }-8=\pm 36\\ \Rightarrow \mathrm{\lambda }-2\mathrm{\mu }+22=0.......\left(2\right)\\ \mathrm{and} \mathrm{\lambda }-2\mathrm{\mu }-14=0 ......\left(3\right)\end{array}$
Solving 1 & 3 $(\mathrm{\lambda },\mathrm{\mu })=\left(\frac{42}{9},\frac{-42}{9}\right)$
Solving 1 & 2 $(\mathrm{\lambda },\mathrm{\mu })=\left(\frac{78}{9},\frac{138}{9}\right)$
The sum of all the possible values of $\mathrm{\lambda }+\mathrm{\mu }=24$