Call one pawn A and the other B. Now, owing to that optional first move,
either pawn may make either 5 or 6 moves in reaching the eighth square.
There are, therefore, four cases to be considered: (1) A 6 moves and B 6
moves; (2) A 6 moves and B 5 moves; (3) A 5 moves and B 6 moves; (4) A 5
moves and B 5 moves. In case (1) there are 12 moves, and we may select
any 6 of these for A. Therefore 7×8×9×10×11×12 divided by 1×2×3×4×5×6
gives us the number of variations for this case--that is, 924. Similarly
for case (2), 6 selections out of 11 will be 462; in case (3), 5
selections out of 11 will also be 462; and in case (4), 5 selections out
of 10 will be 252. Add these four numbers together and we get 2,100,
which is the correct number of different ways in which the pawns may
advance under the conditions. (See No. 270, on p. 204.)