If three square are chosen at random on a chess Board, show that chance that they should be in a diagonal line 7/744.

Options

A. `(7)/(744)`
B. `(5)/(744)`
C. `(7)/(544)`
D. `(11)/(744)`

Answer

A

Explanation

We can choose three squares in a diagonal line parallel to BD in the `Delta ABD`. Clearly, three squares in `Delta ABD` and in a diagonal line parallel to BD can be chosen in

`.^(3)C_(3)+ .^(4)C_(3)+.^(5)C_(3)+ .^(6)C_(3)+ .^(7)C_(3)+ .^(8)C_(3)` ways
Similarly, in `Delta BCD`, the squares can be chosen parallel to BD in an equal number of ways. Hence, the total number of ways in which three squares can be chosen in a diagonal line parallel to BD is
`2(.^(3)C_(3)+.^(4)C_(3)+ .^(5)C_(3)+ .^(6)C_(3)+ .^(7)C_(3))+ .^(8)C_(3)` [`because` BD is common to both the triangles]
Similarly, squares can be chosen in a diagonal line parallel to AC and hence the total number of favourable ways
`= 4(.^(3)C_(3)+ .^(4)C_(3)+ .^(5)C_(3)+ .^(6)C_(3)+ .^(7)C_(3))+2 .^(8)C_(3)=392`
Hence, the required probability `=(392)/(.^(64)C_(3))`
`=(392xx6)/(64.63.62)=(7)/(744)`