$8$-digit numbers are formed using the digits $1, 1, 2, 2, 2, 3, 4, 4$. The number of such numbers in which the odd digits do not occupy odd places is:

If $x, y$ and $r$ are positive integers, then $^x{C_r} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^y{C_r} = $

The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

$A$ question paper had ten questions. Each question could only be answered as True $(T)$ or False $(F)$. Each candidate answered all the questions. Yet,no two candidates wrote the answer in an identical sequence. How many different sequences of answers are possible?

How many $3$-letter words can be formed from the letters of the word '$EXPERTISE$',if repetition of letters is not allowed?

In an examination hall,there are $4$ rows of chairs. Each row has $8$ chairs,$1$ behind the other. There are $2$ classes sitting for the examination with $16$ students in each class. It is desired that in each row,all students belong to the same class and that no $2$ adjacent rows are allotted to the same class. In how many ways can these $32$ students be seated?