Find $r$ if $^{5}P_{r} = ^{6}P_{r-1}$.

The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ which are divisible by $4$,when the repetition of any digit is not allowed,is:

Find the number of arrangements of the letters of the word $BANANA$ in which the two $N$'s do not appear together.

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 the digits in the unit's place of all the $4-$ digit numbers formed by using the digits $3, 4, 5,$ and $6$,without repetition,is:

The number of integers greater than $6000$ that can be formed by using the digits $0, 5, 6, 7, 8,$ and $9$ without repetition is