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(Q). In how many ways two kings one black & one white can be placed on a 8×8 chess board such that they are not on adjacent squares?
Solution:-

Here we have following cases for placing the 1ˢᵗ king.
Case(i): Lets 1ˢᵗ king be placed at one of the 4 corners of chess board ⟹ ⁴C₁ = 4 ways

then 2ⁿᵈ king can be placed at 64 - 4 = 60 square in 60 ways.place of king & its adjacent 3 squares

So number of ways in this case is 4×60 = 240

case(ii): If 1ˢᵗ king is placed at a square on edge other than corner’s square then this can be done in 24 ways

then 2ⁿᵈ king can be placed at 64 - 6 = 58 ways.place of king & its adjacent 5 squares

So number of ways in this case is = 24×58 = 1392
case (iii): If 1ˢᵗ king is placed at an interior square then this can be done in 36 ways

then 2ⁿᵈ king can be placed at 64 - 9 = 55 ways.place of king & its adjacent 8 squares

So number of ways in this case is = 36×55 = 1980
So total number of ways = 240 + 1392 + 1980
                                            = 3612        Answer