Question

Seven different lectures are to be delivered in $7$ periods of a class on a particular day. Out of $7$ lecturers, $A,B$ and $C$ are three of them. If the number of ways in which a routine for the day can be made such that $A$ delivers his lecture before $B$ and $B$ before $C$ is $N,$ then the value of $\left(\frac{N}{120}\right)$ is

Answer 1

Select $3$ places for $A,B,C$ in ${}^7{C}_3$ ways
and arrange them only in $1$ way.
Remaining $4$ places can be arranged in $4!$ ways.
$\therefore$ Total ways $={}^7{C}_3\cdot 4!=35\cdot 24=840$
$\Rightarrow N=840$
$\therefore \left(\frac{N}{120}\right)=\frac{840}{120}=7$