I am in high school and got a question in my textbook which reads:
A baby has nine different toy animals. 5 of them are red and 4 of them are blue. She arranges the toys in a line so that the colours are arranged symmetrically. How many arrangements are possible?
This is how I reasoned: to fulfil the symmetrical requirement, the middle toy must be red. This leaves 8 toys, 4 on each side of the middle red toy - 4 of these are red, and 4 are blue. To calculate the permutations of the remaining 8, keeping the symmetrical requirement in mind, we calculate 4!.
Therefore, the total number of arrangements is $4! + 1 = 25$
This was not the correct answer - the correct answer is $6*5!*4! = 17280$
Could someone please explain where my thinking fails me? Thank you!