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Fischer Random Chess is also called Chess960 because the number of different initial positions is 960.
The number of possible games at the end of the n-th plie is the sum of all possible games on all 960 boards with a different initial position.
The number of possible first moves for white depends on the positions of the knights. If they are on a1 and/or h1 the number of possible moves reduces from 20 to 18 or 19.
On the 960 boards there are 240 boards with a knight on a1. Looking more closely at the positions of the second knight on these 240 boards reveals that 36 knights can be found on b1, d1, f1 and h1 and 32 knights can be found on c1, e1 and g1, something that can be proved with some simple combinatorics.
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