Question

In an examination hall, there are four rows of chairs. Each row has $8$ chairs one behind the other. There are two 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 two adjacent rows are allotted to the same class. If the $32$ students can be seated in $m$ ways, the value of $\frac{m}{(16!{)}^2}$ is:

Answer 1

There are four rows of chairs (say I, II, III, IV) consisting of $8$ chairs each.

It is desired that in each row, all students belong to the same class and no two adjacent rows are alloted to same class.

Therefore, one class can be seated in either I and III or in II and IV, that is in $2$ ways.

Now, $16$ students of this class can be arranged in $16$ chairs in ${}^{16}{P}_{16}=16!$ ways. $16$ students of other class can be arranged in 16 chairs in ${}^{16}{P}_{16}=16!$ ways.
Therefore, total number of ways
$=2\times 6!\times 16!=m$

Hence, $\frac{m}{(16!{)}^2}=2$.