Why not Chess5040?

I am sure that I am misunderstanding something either about Chess960 rules or mathematical enumeration. My current understanding makes me conclude that Chess960 should really be called Chess5040.

The claim is that there are 960 possible combinations/permutations of ways to order the eight primary chess pieces (non-pawns).

But here is my calculation:

We have eight pieces in the first row, two rooks, two knights, two bishops, one king, and one queen. Hence, we are dealing with distinguishable permutations, and we have n = 8 objects.

These facts yield 8!/(2! * 2! * 2! * 1! * 1!) = 7! = 5040 combinations.

Can anyone identify my misunderstanding?

Fischer decided to eliminate some position, for example those where both bishops are on the same color.
There is also a rule about castling : in the initial position the king must be between the rooks.

The rooks must be on either side of the king.

You can think of it as:

Placing the queen, knights, and bishops,

and then there's always only one configuration for the remaining rook and king.

Additionally, the bishops must be on opposite colours.

Choose Bishops->Choose Queen-> Choose knights-> place remaining king and rooks:

(4*4) * (6) * (5C2)

=960

:)

So you have : 4 choices for the light square bishop
4 choices for the dark square bishop.
Then six choices for the queen
Then place both knights : 5 squares for the first and 4 for the second.

Then castling rules allow only one setup for the King and the rooks.

Total : 4*4*6*5*4=1920
Both knight are identical, so you have only 1920/2=960 positions.

Super cool! Thanks!

aaaaand I was hoping I was done with Math. Dammit, now I have to go to bed with numbers floating before my eyes.

Daaaamn youuuu

Thanks wikipedia :)