The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ 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:

All possible $5$-digit numbers,each having $5$ distinct digits,are formed using the digits $\{1, 2, 3, 5, 6, 8\}$. Among them,the number of such numbers which are divisible by $3$ but not by $6$ is:

The number of ways in which a committee of $7$ members can be formed from $6$ teachers,$5$ fathers,and $4$ students in such a way that at least one from each group is included and teachers form the majority among them,is

The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

The number of four-lettered words that can be formed from the letters of the word $MAYANK$ such that both $A$'s are included but never together,is equal to: