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Step-by-step Solution
Step 1: Understand the Problem
We roll four dice simultaneously and use the outcomes (each a number from 1 to 6) to form a 2×2 matrix:
$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}.
$$
We need to find the probability that all four entries are distinct and that the matrix is non-singular (i.e., its determinant is non-zero). The determinant of this 2×2 matrix is
$$ ad - bc. $$
Step 2: Total Number of Possible Outcomes
Each of the four dice can land on any of the 6 faces independently. Hence, the total number of ways to obtain four numbers (and thus form a 2×2 matrix) is:
$$
6^4 = 1296.
$$
Step 3: Count the Ways to Have All Distinct Entries
We must choose 4 distinct values out of the 6 possible faces. The number of ways to choose which 4 distinct numbers appear is
$$
{^6C_4}.
$$
Once those 4 distinct numbers are selected, we can arrange them in the 2×2 matrix in
$$
4!
$$
different ways (since there are 4 positions in the matrix). Therefore, the number of matrices with all distinct entries is
$$
{^6C_4} \times 4!.
$$
Numerically,
$$
{^6C_4} = 15, \quad 4! = 24,
$$
so this product is
$$
15 \times 24 = 360.
$$
Step 4: Subtract the Cases Where the Matrix Is Singular
A 2×2 matrix is singular when
$$
ad - bc = 0 \quad \Longrightarrow \quad ad = bc.
$$
Among the 360 matrices that have distinct entries, count how many satisfy
$$
ad = bc.
$$
After careful enumeration (or systematic checking of all cases with distinct a, b, c, d from {1, 2, 3, 4, 5, 6}), there are 16 such singular cases. Thus, the number of non-singular matrices with distinct entries is:
$$
360 - 16 = 344.
$$
Step 5: Compute the Required Probability
The probability that the randomly formed 2×2 matrix has all distinct entries and is non-singular is the ratio of favorable cases to total cases:
$$
\text{Probability}
= \frac{344}{6^4}
= \frac{344}{1296}
= \frac{43}{162}.
$$
Hence, the answer is
$$
\frac{43}{162}.
$$
