To solve the problem, we need to find the probability that the determinant of matrix \( A \), defined as \( f(A) = \text{det}(A) = ad - bc \), equals 15, where \( a, b, c, d \) can take values from the set \{ -3, -2, -1, 0, 1, 2, 3 \}.
### Step-by-Step Solution:
1. **Understanding the Determinant**:
The determinant of the matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[
f(A) = ad - bc
\]
We need to find the combinations of \( a, b, c, d \) such that \( ad - bc = 15 \).
2. **Finding Maximum and Minimum Values**:
- The maximum value of \( ad \) occurs when both \( a \) and \( d \) are at their maximum, which is \( 3 \times 3 = 9 \).
- The minimum value of \( bc \) occurs when both \( b \) and \( c \) are at their minimum, which is \( -3 \times -3 = 9 \).
- Therefore, the maximum value of \( ad - bc \) is \( 9 - (-9) = 18 \) and the minimum value is \( -9 - 9 = -18 \).
3. **Setting Up the Equation**:
To satisfy \( ad - bc = 15 \), we can rearrange it to:
\[
ad = 15 + bc
\]
This implies that \( bc \) must be less than or equal to \( 9 \) since \( ad \) cannot exceed \( 9 \).
4. **Finding Combinations**:
We need to find pairs \( (a, d) \) and \( (b, c) \) such that:
- \( ad = 9 \) and \( bc = -6 \) (because \( 9 - (-6) = 15 \))
- \( ad = 6 \) and \( bc = -9 \) (because \( 6 - (-9) = 15 \))
5. **Calculating Combinations for \( ad = 9 \)**:
- The pairs \( (a, d) \) that give \( ad = 9 \) are:
- \( (3, 3) \)
- \( (-3, -3) \)
- Each of these pairs can be arranged in \( 2 \) ways (e.g., \( (3, 3) \) and \( (3, 3) \) is the same, but \( (-3, -3) \) is also the same).
- For \( bc = -6 \), the pairs are:
- \( (2, -3), (-3, 2), (3, -2), (-2, 3) \)
- This gives \( 4 \) combinations for \( (b, c) \).
6. **Calculating Total Combinations for \( ad = 6 \)**:
- The pairs \( (a, d) \) that give \( ad = 6 \) are:
- \( (2, 3), (3, 2), (-2, -3), (-3, -2) \)
- This gives \( 4 \) combinations.
- For \( bc = -9 \), the pairs are:
- \( (3, -3), (-3, 3) \)
- This gives \( 2 \) combinations.
7. **Calculating Total Ways**:
- For \( ad = 9 \) and \( bc = -6 \): \( 2 \times 4 = 8 \) ways.
- For \( ad = 6 \) and \( bc = -9 \): \( 4 \times 2 = 8 \) ways.
- Total ways to achieve \( f(A) = 15 \) is \( 8 + 8 = 16 \).
8. **Calculating Total Possible Outcomes**:
Since \( a, b, c, d \) can each take \( 7 \) values, the total number of combinations is:
\[
7^4 = 2401
\]
9. **Calculating Probability**:
The probability that \( f(A) = 15 \) is:
\[
P(f(A) = 15) = \frac{16}{2401}
\]
### Final Answer:
The probability that \( f(A) = 15 \) is \( \frac{16}{2401} \).
