The number of ways in which the letters of the word $ARRANGE$ can be permuted such that the $R$'s occur together is:

How many words can be formed using the letters of the word $ARRANGE$ such that the consonants appear in alphabetical order?

If the letters of the word "$CAPITAL$" are permuted in all possible ways and the words are arranged in a dictionary order,then the rank of the word "$CAPITAL$" is

For $1 \leq r \leq n$,the value of $\frac{1}{r+1}\left\{{ }^n P_{r+1}-{ }^{(n-1)} P_{r+1}\right\}$ is equal to

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

There are $11$ distinct books on a shelf,consisting of $5$ Mathematics,$4$ Physics,and $2$ Chemistry books. In how many ways can they be arranged such that books of the same subject are always kept together?