$(n - r + 1) \times ^nP_{r - 1} = \dots$

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

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

The total number of words that can be formed using the letters $a, b, c, d, e, f$ taken $3$ at a time such that each word contains at least one vowel is:

How many $5$-digit numbers can be formed using the digits $1, 2, 3, 4, 5$ such that the number is divisible by $4$?

How many words,with or without meaning,can be formed from the letters of the word $MONDAY$,assuming that no letter is repeated,if all letters are used at a time?