The White pawns may be arranged in 40,320 ways, the White rooks in 2
ways, the bishops in 2 ways, and the knights in 2 ways. Multiply these
numbers together, and we find that the White pieces may be placed in
322,560 different ways. The Black pieces may, of course, be placed in
the same number of ways. Therefore the men may be set up in 322,560 ×
322,560 = 104,044,953,600 ways. But the point that nearly everybody
overlooks is that the board may be placed in two different ways for
every arrangement. Therefore the answer is doubled, and is
208,089,907,200 different ways.