1. Key Concepts and Formulas

• Probability Formula: The probability of an event $E$ is given by the ratio of the number of favorable outcomes to the total number of possible outcomes: $P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$
• Combinatorics (Permutations): When selecting $k$ distinct items from a set of $n$ distinct items and arranging them in a specific order, we use permutations, denoted as $P(n,k)$ or $_nP_k$: $P(n,k) = \frac{n!}{(n-k)!}$
• Determinant of a $2 \times 2$ Matrix: For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, its determinant is $\det(A) = ad - bc$.
• Non-Singular Matrix: A square matrix is non-singular if its determinant is non-zero. For a $2 \times 2$ matrix, this means $ad - bc \neq 0$, or equivalently, $ad \neq bc$.

2. Step-by-Step Solution

This problem requires us to find the probability of forming a $2 \times 2$ matrix from four dice rolls such that all its entries are distinct and the matrix is non-singular.

Step 1: Calculate the Total Number of Possible Outcomes

• What we are doing: Determining the total size of our sample space, which is the total number of unique $2 \times 2$ matrices that can be formed from four dice rolls.
• Why: This will be the denominator in our probability calculation.
• Explanation: When four dice are thrown, each die can land on any integer from 1 to 6. If we record these four numbers as the entries $a_{11}, a_{12}, a_{21}, a_{22}$ of a $2 \times 2$ matrix, then each position can be filled in 6 ways, independently.
• Calculation: Number of choices for $a_{11} = 6$ Number of choices for $a_{12} = 6$ Number of choices for $a_{21} = 6$ Number of choices for $a_{22} = 6$ Total number of possible $2 \times 2$ matrices = $6 \times 6 \times 6 \times 6 = 6^4$. $N(\text{Total Outcomes}) = 6^4 = 1296$

Step 2: Calculate the Number of Matrices with All Different Entries

• What we are doing: Identifying the number of matrices that satisfy the first condition: all entries must be distinct.
• Why: This narrows down our sample space to only those matrices that meet the first requirement for being a "favorable outcome."
• Explanation: We need to choose 4 distinct numbers from the 6 possible outcomes of a die roll ($\{1, 2, 3, 4, 5, 6\}$) and arrange them into the four positions of the $2 \times 2$ matrix. This is a permutation problem.
• Calculation: The number of ways to choose 4 distinct numbers from 6 and arrange them in 4 specific positions is given by the permutation formula $P(n,k) = \frac{n!}{(n-k)!}$, where $n=6$ and $k=4$. $N(\text{Matrices with Distinct Entries}) = P(6,4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = 6 \times 5 \times 4 \times 3 = 360$ So, there are 360 matrices where all entries are distinct.

Step 3: Calculate the Number of Non-Singular Matrices (among those with distinct entries)

• What we are doing: From the 360 matrices with distinct entries, we now apply the second condition: the matrix must be non-singular.

• Why: A matrix is non-singular if its determinant is non-zero. For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, this means $ad - bc \neq 0$, or $ad \neq bc$. It's often easier to count the opposite (singular matrices, where $ad=bc$) and subtract this from the total number of distinct-entry matrices.

• Explanation: We are looking for four distinct numbers $a, b, c, d$ from the set $\{1, 2, 3, 4, 5, 6\}$ such that their product $ad$ equals their product $bc$.

• Systematic Search for Singular Matrices with Distinct Entries ($ad=bc$): We list all possible products of two distinct numbers from $\{1, 2, 3, 4, 5, 6\}$ and look for pairs of products that are equal, ensuring that all four numbers involved are distinct.

  1. Products of two distinct numbers: $1 \times 2 = 2$ $1 \times 3 = 3$ $1 \times 4 = 4$ $1 \times 5 = 5$ $1 \times 6 = 6$ $2 \times 3 = 6$ $2 \times 4 = 8$ $2 \times 5 = 10$ $2 \times 6 = 12$ $3 \times 4 = 12$ $3 \times 5 = 15$ $3 \times 6 = 18$ $4 \times 5 = 20$ $4 \times 6 = 24$ $5 \times 6 = 30$

  2. Identify cases where $ad = bc$ with $a,b,c,d$ all distinct:

     • Case A: Product = 6 We have two pairs of distinct numbers whose product is 6: $\{1,6\}$ and $\{2,3\}$. The four numbers involved are $\{1,2,3,6\}$, which are all distinct.

       • Subcase 1: Let $\{a,d\} = \{1,6\}$ and $\{b,c\} = \{2,3\}$.
         • The pair $(a,d)$ can be $(1,6)$ or $(6,1)$ (2 ways).
         • The pair $(b,c)$ can be $(2,3)$ or $(3,2)$ (2 ways).
         • This gives $2 \times 2 = 4$ matrices (e.g., $\begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix}$, $\begin{pmatrix} 6 & 2 \\ 3 & 1 \end{pmatrix}$, etc.).
       • Subcase 2: Let $\{a,d\} = \{2,3\}$ and $\{b,c\} = \{1,6\}$.
         • The pair $(a,d)$ can be $(2,3)$ or $(3,2)$ (2 ways).
         • The pair $(b,c)$ can be $(1,6)$ or $(6,1)$ (2 ways).
         • This also gives $2 \times 2 = 4$ matrices. Total singular matrices for this set of numbers $\{1,2,3,6\}$ is $4 + 4 = 8$.

     • Case B: Product = 12 We have two pairs of distinct numbers whose product is 12: $\{2,6\}$ and $\{3,4\}$. The four numbers involved are $\{2,3,4,6\}$, which are all distinct.

       • Subcase 1: Let $\{a,d\} = \{2,6\}$ and $\{b,c\} = \{3,4\}$.
         • The pair $(a,d)$ can be $(2,6)$ or $(6,2)$ (2 ways).
         • The pair $(b,c)$ can be $(3,4)$ or $(4,3)$ (2 ways).
         • This gives $2 \times 2 = 4$ matrices.
       • Subcase 2: Let $\{a,d\} = \{3,4\}$ and $\{b,c\} = \{2,6\}$.
         • The pair $(a,d)$ can be $(3,4)$ or $(4,3)$ (2 ways).
         • The pair $(b,c)$ can be $(2,6)$ or $(6,2)$ (2 ways).
         • This also gives $2 \times 2 = 4$ matrices. Total singular matrices for this set of numbers $\{2,3,4,6\}$ is $4 + 4 = 8$.

     • No other common products yield four distinct numbers. For example, while $1 \times 4 = 4$ and $2 \times 2 = 4$, the numbers $\{1,4,2,2\}$ are not distinct.

• Total Number of Singular Matrices with Distinct Entries: Adding the counts from Case A and Case B: $8 + 8 = 16$.

• Number of Favorable Outcomes (Distinct and Non-Singular): This is the total number of matrices with distinct entries minus those that are also singular. $N(\text{Favorable Outcomes}) = N(\text{Distinct Entries}) - N(\text{Singular and Distinct Entries})$ $N(\text{Favorable Outcomes}) = 360 - 16 = 344$

Step 4: Calculate the Probability

• What we are doing: Applying the fundamental probability formula using the results from Step 1 and Step 3.
• Why: This gives us the final desired probability.
• Calculation: $P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{344}{1296}$ To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor. Both are divisible by 8: $344 \div 8 = 43$ $1296 \div 8 = 162$ $P(\text{Event}) = \frac{43}{162}$

3. Common Mistakes & Tips

• Distinctness is Key: When counting matrices with distinct entries, ensure that all four chosen numbers are unique, and then consider their arrangements. This is a permutation, not just a combination.
• Systematic Enumeration for $ad=bc$: When searching for singular matrices ($ad=bc$), be methodical. List all possible products of two distinct numbers and carefully identify common products where all four numbers involved ($a, b, c, d$) are distinct. Avoid double-counting or missing cases.
• Permutations of Pairs: Remember that if $ad=bc$, then for a given set of four distinct numbers, say $\{x_1, x_2, x_3, x_4\}$, if $x_1x_2 = x_3x_4$, then $(a,d)$ can be $(x_1, x_2)$ or $(x_2, x_1)$, and similarly for $(b,c)$. Also, the roles of $\{a,d\}$ and $\{b,c\}$ can be swapped.

4. Summary

To solve this probability problem, we first determined the total number of possible $2 \times 2$ matrices that can be formed from four dice rolls ($6^4$). Next, we calculated the number of matrices where all entries are distinct using permutations ($P(6,4)$). From this set of distinct-entry matrices, we then identified and subtracted the matrices that are singular (i.e., $ad=bc$). This was done by systematically finding pairs of distinct numbers whose products are equal, ensuring all four numbers involved were unique. Finally, the probability was calculated by dividing the number of favorable outcomes (distinct and non-singular matrices) by the total number of possible outcomes and simplifying the fraction.

The final probability is $\frac{43}{162}$.

5. Final Answer

The final answer is \boxed{\text{{43} \over {162}}}, which corresponds to option (A).