Question

The number of ways in which 4 squares can be chosen at random on a chess board such that they lie on a diagonal line?

Answer 1

Answer:

THE ANSWER FOR THIS QUESTION IS : 364

Explanation:

There are 64 squares in a regular chessboard. therefore we can draw 15 diagonals in one direction and other 15 diagonals in the other direction.

Here we have to select diagonals with 4 or more squares. We can clearly see that

Diagonal 1 = 1 square

Diagonal 2 = 2 squares

Diagonal 3 = 3 squares

Diagonal 4 = 4 squares

Diagonal 5 = 5 squares

Diagonal 6 = 6 squares

Diagonal 7= 7 squares

Diagonal 8 = 8 squares

Diagonal 9 = 7 squares

Diagonal 10 = 6 squares

Diagonal 11 = 5 squares

Diagonal 12 = 4 squares

Diagonal 13 = 3 squares

Diagonal 14 = 2 squares

Diagonal 15 = 1 square

We need to select diagonals with 4 or more squares only

therefore,

D4+D5+D6+D7+D8+D9+D10+D11+D12

= 4C4 + 5C4 + 6C4 + 7C4 + 8C4 + 7C4 + 6C4 + 5C4 + 4C4

= 1 + 5 + 15 + 35 + 70 + 35 + 15 + 5 + 1

= 182

Similar for the other one direction

182 + 182

= 364

The answer for this question is 364