Question

Number of ways of choosing four squares on the Chess board in such a way that all squares are on one diagonal line is

• A. $504$
• B. $182$
• C. $364$
• D. $252$

Answer 1

The correct option is C $364$

Let consider $△ABC$.
Number of ways of selecting four squares along the lines $A_1{C}_1,A_2{C}_2,A_3{C}_3,A_4{C}_4,AC,A_5{C}_5,A_6{C}_6,A_7{C}_7$ and $A_8{C}_8$ is ${}^4{C}_4,{}^5{C}_4,{}^6{C}_4,{}^7{C}_4,{}^8{C}_4,{}^7{C}_4,{}^6{C}_4,{}^5,C_4$ and ${}^4{C}_4$ repectively

$\Rightarrow 2[{}^4{C}_4,{}^5{C}_4,{}^6{C}_4,{}^7{C}_4]+{}^8{C}_4$
In another case take all the diagonal parallel to to line $BD$
Numbe rof ways are : $2[{}^4{C}_4+{}^5{C}_4+{}^6{C}_4+{}^7{C}_4]+{}^8{C}_4$
Hence, required number of ways : $=2\left[2\right[{}^4{C}_4+{}^5{C}_4+{}^6{C}_4+{}^7{C}_4]+{}^8{C}_4]$
$=2\left[2\right[1+5+15+35]+70]$
$=364$