Question

Number of ways in which $5A^{\prime }s$ and $6B^{\prime }s$ can be arranged in a row which reads the same backwards and forwards, is $N$ then find the value of $N/2$.

Answer 1

$AAAAA|BBBBB$
Since word reads the same backwards and forwards, the middle digit must be $A$.
$\underset{\times \times \times \times \times \downarrow \times \times \times \times \times }{M}$
so that even number of $A^{\prime }s$ and $B^{\prime }s$ are available for arrangement about middle position $M$ in the above figure.

Take $AABBB$ on one side of $M(6^{th}$ place) and then their image about $M$ in a unique way
$\therefore$ Number of ways $N=\frac{5!}{2!\cdot 3!}=10$.

then the value of $N/2$ is $5$.