Ten persons,amongst whom are $A, B$ and $C$,are to speak at a function. The number of ways in which it can be done if $A$ wants to speak before $B$ and $B$ wants to speak before $C$ is:

On the occasion of the Deepawali festival,each student of a class sends greeting cards to every other student. If there are $20$ students in the class,then the total number of greeting cards exchanged by the students is:

The letters of the word '$MANKIND$' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word '$MANKIND$' is $.....$

The sum of all $4$-digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

Seven scientists $S_1, S_2, \ldots, S_7$ are invited to deliver one lecture each in a conference. The number of ways all the seven lectures can be arranged such that the lecture of $S_1$ is prior to that of $S_3$ and the lecture of $S_3$ is prior to that of $S_7$ is

Evaluate $7!$