Question

The staff of a department consist of a manager, an officer and $10$ clerks. A committee of $4$ to be selected from this department. Find the number of ways in which this can be done so as to always include:
i) the Manager
ii) the Manager but not the officer.

Answer 1

Total number of people $=1manager+1officer+10clerks=12$

$i)$ Number of ways selecting the manager $=1$

Hence, the possible ways of selecting $4$ out of $12$ including the manager ${=}^{11}{c}_3\times 1$

$=\frac{11\times 10\times 9}{3!}$

$=165$

$i)$ Number of ways selecting the manager $=1.$

Number of ways selecting the officer $=1.$

Hence, the possible ways of selecting $4$ out of $12$ including the manager ${=}^{10}{c}_2\times 1\times 1$

$=\frac{11\times 10}{2!}$

$=55$