The number of ways of getting a sum of $16$ when throwing a die four times is:

In the product $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{100})$,when written in ascending powers of $x$,the highest exponent of $x$ is . . . . . . .

There are $15$ stations on a train route and the train has to be stopped at exactly $5$ stations among these $15$ stations. If it stops at at least two consecutive stations,then the number of ways in which the train can be stopped is

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:

Consider the set $A = \{1, 2, 3, \ldots, 30\}$. The number of ways in which one can choose three distinct numbers from $A$ such that the product of the chosen numbers is divisible by $9$ is:

$A$ certain question paper contains three parts $A, B, C$ with four questions in part $A$,five questions in part $B$,and six questions in part $C$. $A$ student is required to answer seven questions,choosing at least two questions from each part. The total number of different ways a student can choose his seven questions for answering is: