Mathematics - Probability Question with Solution | TestHub

$\begin{array}{rl}& 3\text{ engineering }+1\text{ doctor }+8\text{ Prof }\to {}^4{\mathrm{C}}_3\cdot {}^2{\mathrm{C}}_1\cdot {}^{10}{\mathrm{C}}_8\\ & =360\\ & 3\text{ engineering }+2\text{ doctors }+7\text{ Prof }\to {}^4{\mathrm{C}}_3\cdot {}^2{\mathrm{C}}_2\cdot {}^{10}{\mathrm{C}}_7\\ & =480\\ & 4\text{ engineering }+1\text{ doctor }+7\text{ Prof }\to {}^4{\mathrm{C}}_4\cdot {}^2{\mathrm{C}}_1\cdot {}^{10}{\mathrm{C}}_7\\ & =240\\ & 4\text{ engineering }+2\text{ doctors }+6\text{ Prof }\to {}^4{\mathrm{C}}_4\cdot {}^2{\mathrm{C}}_2\cdot {}^{10}{\mathrm{C}}_6\\ & =210\end{array}$
Total $=1290$
Req. probability $=\frac{1290}{{}^{16}{\mathrm{C}}_{12}}=\frac{1290}{1820}=\frac{129}{182}$