$A$ question paper contains $4$ questions,each having $4$ alternative answers. The number of ways that a candidate can answer one or more questions is

Number of $4-$digit numbers (the repetition of digits is allowed) which are made using the digits $1, 2, 3$ and $5$,and are divisible by $15$,is equal to $............$.

The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by

For a set of $5$ true or false questions,no student has written all the correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

$A$ five-digit number divisible by $3$ has to be formed using the numerals $0, 1, 2, 3, 4,$ and $5$ without repetition. The total number of ways in which this can be done is:

In an examination,there are $3$ multiple-choice questions,and each question has $4$ choices. The number of ways in which a student can fail to get all answers correct is: