Question

Number of ways in which 5 A's and 6 B's can be arranged in a row which reads the same backwards and forwards is

Answer 1

AAAAA | BBBBBB
Since word reads the same backwards and forwards, the middle digit must be A.
M
$\times \times \times \times \times \downarrow \times \times \times \times \times$
so that even number of A's and B's are available for arrangement about middle position M in the above figure.
Take AABBB on one side of M (6th place) and then their image about M in a unique way
$\therefore$ The total number of required ways $=\frac{5!}{2!.3!}=10$