MAT Exam 2020 Paper

Concept:This is a problem of selection followed by arrangement. First choose the required persons, then arrange them in the line.Explanation:Since $P_1$ must always occur, $P_1$ is fixed as one of the 5 selected persons.Since $P_4$ and $P_5$ must not occur, both are removed from consideration.Available persons for selection $= 10 - 1 - 2 = 7$.We already have $P_1$, so we need to select 4 more persons from these 7 available persons.Number of ways to select 4 persons from 7 is ${}^{7}C_{4}$.After selecting 5 persons, they can be arranged in a line in $5!$ ways.Therefore, total number of possible arrangements $= {}^{7}C_{4} \times 5!$.Answer:The required number of arrangements is ${}^{7}C_{4} \times 5!$, which matches option A.