The number of ways of selecting two 1 × 1 squares from a chess board such that they

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have a common vertex is 98

have a common side is 112

none of these

neither have a common vertex nor have a common side is 1806

answer is A, B, C.

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Detailed Solution

Consider square of 2 × 2, in which we have 2 pairs of
squares which have common vertex.
We have such 7 × 7squares of size 2 × 2.
So, number of ways of selecting two squares having
one vertex common is 7 × 7 × 2 = 98
In each row or column we have 7 pairs of squares
having one side common.
So, number of ways of selecting two squares having
one side common is 7 × 8× 2 = 112.
Therefore, number of ways of selecting two squares
such that they neither have a common vertex nor
have a common side is $^{64} C_{2} - 98 - 112 = 1806$