Ten students are seated at random in a row. The probability that two particular students are not seated side by side is

The number of $6$-letter words,with or without meaning,that can be formed using the letters of the word $\text{MATHS}$ such that any letter that appears in the word must appear at least twice,is $...$

The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

The number of different words that can be formed using all the letters of the word $APPLICATION$ such that no two vowels come together is -

The sum of all possible numbers that can be formed by using the digits $2, 3, 5, 7$ without repetition of digits is

If $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$,then $\sum_{r=0}^n \frac{r}{^nC_r}$ equals