In how many ways 4 square are can be chosen on a chess-board, such that all the squares lie in a diagonal line.
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Sol. Let us consider the Δ ABC. Number of ways in which 4 selected squares are along the lines
A 4 C 4 , A 3 C 3 , A 2 , C 2 , A 1 C 1 and AC are 4 C 4 , 5 C 4 , 6 C 4 and 8 C 4 respectively.
Similarly, in Δ ACB, number of ways in which 4 selected squares are along the diagonal line parallel to AC are 4 C 4 , 5 C 4 , 6 C 4 , 7 C 4 and 8 C 4 but 8 C 4 triangles occur only once.
Hence the total number of ways in which the 4 selected squares are in a diagonal line parallel to AC are
2( 4 C 4 + 5 C 4 + 6 C 4 + 7 C 4 ) + 8 C 4
Also same is the case of selecting 4 squares on a chess-board. Such that the 4 squares are in a diagonal line = 2 [2( 4 C 4 + 5 C 4 + 6 C 4 + 7 C 4 ) + 8 C 4 ]
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