The letters of the word $ARRANGE$ are used to form different words. All these words are arranged in dictionary order. What is the rank of the word $ARRANGE$ in the dictionary?

The number of ways of arranging the letters of the word $BANANA$ so that the two $N$s do not come together is:

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:

How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 5, 7$ without repetition? If all such numbers are arranged in ascending order,what is the rank of the number $70513$?

In how many ways can $5$ balls be placed in $4$ tins if any number of balls can be placed in any tin?

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