Question

how many ways can 12 identical white and 12 identical black pawns in the black squares in a 8*8 chess board?​

Answer 1

Step-by-step explanation:

You can choose 1212 places out of 3232 for the white pawns, then 1212 places out of the remaining 2020 for the black pawns.

(3212)⋅(2012)=32!12!⋅20!⋅20!12!⋅8!=32!12!⋅12!⋅8!

(3212)⋅(2012)=32!12!⋅20!⋅20!12!⋅8!=32!12!⋅12!⋅8!

Note that 12+12+8=3212+12+8=32, so this is the formula for splitting the squares between three collections - white pawns, black pawns and empty squares. The formula has to be the same whichever way you count, which explains the three factorials in the denominator. If you wanted twelve black pawns and eight empty squares (the rest being white pawns) you would expect the same result

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