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If four squares are chosen at random on a chess board. If the probability that they lie on a diagonal line is $\mathrm{ \frac{\lambda}{{ }^{64} C_4}}$ , then the value of $\lambda$ must be
Option: 1
$\lambda=1$

Option: 2
$\lambda=365$

Option: 3
$\lambda=366$

Option: 4
$\lambda=364$

Answers (1)

best_answer

The number of ways $=^{64}C_4$

The chess board can be divided into two parts by a diagonal line $\mathrm{B D}$ . Now, if we begin to select four squares from the diagonal $\mathrm{P_1 Q_1, P_2 Q_2, \ldots, B D}$ , then we can find number squares selected

$\mathrm{ =2\left({ }^4 C_4+{ }^5 C_4+{ }^6 C_4\right)+{ }^7 C_4 }$

$\mathrm{ =182}$

and similarly number of squares for the diagonals chosen parallel to $\mathrm{A C=182}$
$\mathrm{\therefore}$ Total favourable ways $\mathrm{=364}$

$\therefore$ Required probability $\mathrm{=\frac{364}{{ }^{64} C_4}}$

$\mathrm{\therefore \quad \lambda=364}$

Hence option 4 is correct.

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jitender.kumar

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