The number of distinct real values of $\lambda$, for which the vectors $-{\lambda }^2\widehat{i}+\widehat{j}+\widehat{k},\widehat{i}-{\lambda }^2\widehat{j}+\widehat{k}\text{ and }\widehat{i}+\widehat{j}-{\lambda }^2\widehat{k}$ are coplanar is 

We know that three vectors are coplanar if their scalar triple product is zero. Thus, $\left|\begin{array}{ccc}-{\lambda }^2& 1& 1\\ 1& -{\lambda }^2& 1\\ 1& 1& -{\lambda }^2\end{array}\right|=0$

$\Rightarrow R_1\to R_1+R_2+R_3\Rightarrow \lambda =\pm \sqrt{2}$

Hence, two real solutions.