If `(x-5)^(x^(2)-5x+6)=1` , then (A)sum of all integral solution is 11 (B)sum of all integral solution is 16 (C)product of all integral solution is 144 (D)product of all integral solutions is 30

Options

A. sum of all integral solution is 11
B. sum of all integral solution is 16
C. product of all integral solution is 144
D. product of all integral solutions is 30

Explanation

To solve the equation \((x-5)^{(x^2-5x+6)} = 1\), we need to analyze when an expression raised to a power equals 1. This can occur under certain conditions:

1. The base is 1: \(x - 5 = 1\)
2. The base is -1 and the exponent is even: \(x - 5 = -1\) and \(x^2 - 5x + 6\) is even.
3. The exponent is 0 (and the base is not 0): \(x^2 - 5x + 6 = 0\)

### Step 1: Solve \(x - 5 = 1\)
\[ x - 5 = 1 \implies x = 6 \]

### Step 2: Solve \(x - 5 = -1\)
\[ x - 5 = -1 \implies x = 4 \]
Now we need to check if the exponent is even:
\[ x^2 - 5x + 6 = 4^2 - 5 \cdot 4 + 6 = 16 - 20 + 6 = 2 \quad (\text{even}) \]
Thus, \(x = 4\) is a valid solution.

### Step 3: Solve \(x^2 - 5x + 6 = 0\)
Factoring the quadratic:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]
This gives us:
\[ x = 2 \quad \text{and} \quad x = 3 \]

### Step 4: Compile all solutions
The integral solutions we found are:
- \(x = 2\)
- \(x = 3\)
- \(x = 4\)
- \(x = 6\)

### Step 5: Calculate the sum of all integral solutions
\[ \text{Sum} = 2 + 3 + 4 + 6 = 15 \]

### Step 6: Calculate the product of all integral solutions
\[ \text{Product} = 2 \times 3 \times 4 \times 6 \]
Calculating step-by-step:
\[ 2 \times 3 = 6 \]
\[ 6 \times 4 = 24 \]
\[ 24 \times 6 = 144 \]

### Final Results
- The sum of all integral solutions is \(15\).
- The product of all integral solutions is \(144\).

### Conclusion
The correct answer is:
- The product of all integral solutions is \(144\) (Option C).