Four dice thrown, numbers recorded in a 22 matrix.

Question

Four dice thrown, numbers recorded in a $2	imes2$ matrix. Find the probability the matrix has all different entries and is non-singular.

Accepted Answer

$n(S)=6^4$. Counting singular matrices ($ad=bc$) with all distinct entries gives 16 cases (via cross-ratio analysis over $\{1,\dots,6\}$). Number with all-distinct entries: ${}^6C_4\times4! = 360$.

$$P = \frac{360-16}{1296} = \frac{344}{1296}$$

*(The source's final simplified value is $43/162$.)*

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