Question

If four squares are chosen at random on a chess board, The probability that they lie on a diagonal line is $=\frac{91k}{{{}^{64}C}_4}$.Find the value of $k$

Answer 1

$S:$ We can select $4$ square feet of $64$ by ${}^{64}{C}_4$.
Now, for favourable number of cases there are $15$ diagonals in each direction, but only $9$ of them are long enough to contain $4$ squares.
specifically, in each direction, there are two diagonals of length $4$, two of length $5$, two of length $6$, two of length $7$ and one of length $8$.
So number of cases
$=2\left\{{}^4{C}_4{+}^5{C}_4{+}^6{C}_4{+}^7{C}_4\right\}{+}^8{C}_4$
$=2(1+5+15+35)+70=182$
This is along one diagonal same this for another diagonal.
So total is $182+182=364$
$\therefore$ Required probability $=\frac{364}{{}^{64}{C}_4}$
$\therefore$ $k=4$