The number of $4$-digit numbers that are not divisible by $5$ is:

$^{29}C_5 + \sum_{r=0}^{4} {^{(33-r)}C_4} =$

There are $4$ men and $5$ women in Group $A$,and $5$ men and $4$ women in Group $B$. If $4$ persons are selected from each group,then the number of ways of selecting $4$ men and $4$ women is:

$5$ boys and $6$ girls are arranged in all possible ways. Let $X$ denote the number of linear arrangements in which no two boys sit together and $Y$ denote the number of linear arrangements in which no two girls sit together. If $Z$ denotes the number of ways of arranging all of them around a circular table such that no two boys sit together,then $X: Y: Z=$

In an examination,there are $5$ multiple choice questions with $3$ choices each,out of which exactly one is correct. There are $3$ marks for each correct answer,$-2$ marks for each wrong answer,and $0$ marks if the question is not attempted. The number of ways a student appearing in the examination can get exactly $5$ marks 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: