• # question_answer There are 10 persons named $A,\ B,.......J$. We have the capacity to accommodate only 5. In how many ways can we arrange them in a line if $A$ is must and $G$ and $H$ must not be included in the team of 5 A) $^{8}{{P}_{5}}$ B) $^{7}{{P}_{5}}$ C) $^{7}{{C}_{3}}(4\ !)$ D) $^{7}{{C}_{3}}(5\ !)$

Out of 10 persons, $A$ is in and $G$ and $H$ are out of the team, so we have to select 4 more from 7 remaining. This can be done in $^{7}{{C}_{4}}$ ways. These 5 persons can be arranged in a line in $5\ !$ ways. Hence the number of possible arrangements is $^{7}{{C}_{4}}.\ 5\ !\ {{=}^{7}}{{C}_{3}}.\,(5\ !)$.