Question

Two variants of a test are suggested to twelve students. In how many ways can the students be placed in two rows so that there should be no identical variants side-by-side and that the students sitting one behind the other should have the same variant?

Answer 1

Consider the seat in the front row and the one behind it in the basic now$.$ So$,$ we choose there pair of seats for each variant$.$

No$:$ of seats paired from $1$ to $6.$

We have $2$ choices for the variant assigned to seat pair $1.$

Then if the variant is also assigned to seat$.$

Pair $3$ or $5,$ the remaining seat pair can be assigned in $3$ different ways$.$

Similarly it is also assigned to seat $4$ and $6.$

$\Rightarrow 2(2.2+4+2)=20$ different assignments

$\Rightarrow 20\cdot {4!}^3=6!{4!}^2$