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I have been given the problem, how many incorrect ways are there of arranging the back row of a chess board? Can anyone help?
Thanks
You mean, how many ways are there of putting the pieces on a chessboard such that they aren't ordered as Rook Knight Bishop Queen King Bishop Knight Rook?
Well, there are 8! = 40320 total arrangements of the pieces.
However, because some of the pieces are identical, we've counted some arrangements twice.
We need to divide by 2 because for every combination we count, we've counted the same one again with the rooks switched, even though that is the same thing.
We need to divide by 2 again for the knights, and again for the bishops. So now the total number of arrangements, without repitition, is 5040.
One of these is the correct arrangement, which means that there are 5039 wrong ways of doing it.
Why did the vector cross the road?
It wanted to be normal.
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