There are $m$ books in black cover and $n$ books in blue cover,and all books are different. The number of ways these $(m+n)$ books can be arranged on a shelf so that all the books in black cover are put side by side is

Evaluate $\frac{n!}{(n-r)!}$ when $n=6$ and $r=2$.

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 number of $4$-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word $INCONSEQUENTIAL$, without repeating any letter, is:

Total number of four-digit odd numbers that can be formed using $0, 1, 2, 3, 5, 7$ (repetition is allowed) is:

The total number of words that can be formed by using all the letters of the word $ALLEN$ such that all vowels occur in alphabetical order is: