The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$ is

The number of permutations of the letters $a_1, a_2, a_3, a_4, a_5$ in which the first letter $a_1$ does not occupy the first position and the second letter $a_2$ does not occupy the second position is:

There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?

Total number of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1, 2, 3, 4, 5$ with repetition is $.......$.

Let $x$ denote the number of ways of arranging $m$ boys and $m$ girls in a row so that no two boys sit together. If $y$ and $z$ denote the number of ways of arranging $m$ boys and $m$ girls in a row and around a circular table respectively so that boys and girls sit alternately,then $x: y: z=$

$A$ test containing $3$ objective-type questions is conducted in a class. Each question has $4$ options and only one option is the correct answer. No two students of the class have answered identically,and no student has written all correct answers. If every student has attempted all the questions,then the maximum possible number of students who have written the test is: