Question

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 12 students be seated?

• A. $2\times 16!\times 16!$
• B. $2\times 15!\times 15!$
• C. $2\times 16!\times 15!$
• D. $16!\times 16!$
• E. None of these

Answer 1

The correct option is A $2\times 16!\times 16!$

There are 4 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 2 adjacent rows are allotted to the same class.
Therefore, 1 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 remaining 16 chairs in ${}^{16}{P}_{16}=16!ways.$
$\therefore$ Total number of ways $=2\times 16!\times 16!$