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

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

All the arrangements,with or without meaning,of the word $FARMER$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $FARMER$ in this list is .... .

If $x$ and $y$ represent the number of arrangements of the letters of the word $ATRAPATRAM$ such that $(i)$ all $A$'s are together and $(ii)$ no two $A$'s are together respectively,then $x+y$ is equal to:

There are $10$ red and $5$ yellow roses of different sizes. If $x$ is the number of garlands that can be formed with all these flowers so that no two yellow roses come together and $y$ is the number of garlands formed with all these flowers so that all the red roses come together,then $\frac{2(x-y)}{10!}=$

If $\frac{{}^{n + 2}C_6}{{}^{n - 2}P_2} = 11$,then $n$ satisfies the equation