$\text{ If the equation of the plane containing the line }x+2y+3z-4=0=2x+y-z+5\text{ and }$perpendicular to the plane  $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\mathrm{\lambda }(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mathrm{\mu }(\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}})\text{ is ax }+\mathrm{by}+\mathrm{cz}=4$, $\text{ then }(\mathrm{a}-\mathrm{b}+\mathrm{c})\text{ is equal to }$

The equation of plane is
$\text{ P: }(x+2y+3z-4)+\lambda (2x+y-z+5)=0$
$\begin{array}{l}\mathrm{P}:(1+2\mathrm{\lambda })\mathrm{x}+(2+\mathrm{\lambda })\mathrm{y}+(3-\mathrm{\lambda })\mathrm{z}-4+5\mathrm{\lambda }=0\\ {\overline{\mathrm{n}}}_1:1+2\mathrm{\lambda },2+\mathrm{\lambda },3-\mathrm{\lambda }\\ {\overline{\mathrm{n}}}_2:\left|\begin{array}{ccc}\widehat{\mathrm{i}}& \widehat{\mathrm{j}}& \widehat{\mathrm{k}}\\ 1& 1& 1\\ 1& -2& 3\end{array}\right|=5\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}-3\widehat{\mathrm{k}}\\ {\overline{\mathrm{n}}}_1\cdot {\overline{\mathrm{n}}}_2=0\Rightarrow 5+10\mathrm{\lambda }-4-2\mathrm{\lambda }-9+3\mathrm{\lambda }=0\Rightarrow \mathrm{\lambda }=\frac{8}{11}\\ \mathrm{P}:\frac{27}{11}\mathrm{x}+\frac{30}{11}\mathrm{y}+\frac{25}{11}\mathrm{z}=\frac{4}{11}\end{array}$
$$\begin{gathered} p:27x+30y+25z=4 \\ a-b+c=27-25+25=22 \end{gathered}$$