Question

Number of ways of selecting pair of black squares in chessboard such that they have exactly one common corner is equal to:

• A. $150$
• B. $56$
• C. $49$
• D. $50$

Answer 1

The correct option is A $150$

let us consider one white corner.

Each vertical and horizontal line will have 4 black squares.

Two black squares beside the white square will have a single corner in common.

Now, put one of them constant, and take other 3 black squares on the same line.

Then, total possible combinations are 4.

Now, doing the same process keeping the other one constant will totally give, $4\times 4$ possibilities.

And, one black square on diagonal and one on vertical line again give $4\times 4$ possibilities.

And, one black square on diagonal and one on horizontal line will give again $4\times 4$ possibilities.

Now, for one white corner we have $4\times 4+4\times 4+4\times 4=16+16+16=48$ possibilities

For, two white squares $2\times 48=96$ possibilities.

Now, for one black corner:

We have to neglect, that black square, because if we consider that, they will have two common corners.

Then, each line will have 3 black squares.

So, now, total possibilities for each black corner $=3\times 3+3\times 3+3\times 3=27$ possibilities

We will have to black squares $=2\times 27=54$ possibilities.

In total we will have $=96+54=150$ possibilities.

Hence, we have $150$ possibilities.