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 the conditions under which an expression raised to a power equals 1. There are three main cases to consider:

1. The base is 1.
2. The base is -1 and the exponent is an even integer.
3. The exponent is 0 (and the base is not 0).

Let's break down the solution step by step:

### Step 1: Analyze the Equation
The equation is \((x-5)^{(x^2-5x+6)} = 1\). We will consider the cases mentioned above.

### Step 2: Case 1 - Base is 1
Set the base equal to 1:
\[ x - 5 = 1 \]
Solving this gives:
\[ x = 6 \]

### Step 3: Case 2 - Base is -1 and Exponent is Even
Set the base equal to -1:
\[ x - 5 = -1 \]
Solving this gives:
\[ x = 4 \]
Now, we need to check if the exponent is even:
\[ x^2 - 5x + 6 = 4^2 - 5 \cdot 4 + 6 = 16 - 20 + 6 = 2 \quad (\text{which is even}) \]
Thus, \(x = 4\) is a valid solution.

### Step 4: Case 3 - Exponent is 0
Set the exponent equal to 0:
\[ x^2 - 5x + 6 = 0 \]
Factoring gives:
\[ (x - 2)(x - 3) = 0 \]
Thus, the solutions are:
\[ x = 2 \quad \text{and} \quad x = 3 \]

### Step 5: Collect All Solutions
The integral solutions we have found are:
\[ x = 2, 3, 4, 6 \]

### Step 6: Calculate the Sum of Solutions
Now, we calculate the sum of all integral solutions:
\[ 2 + 3 + 4 + 6 = 15 \]

### Step 7: Calculate the Product of Solutions
Next, we calculate the product of all integral solutions:
\[ 2 \times 3 \times 4 \times 6 = 144 \]

### Final Answer
The sum of all integral solutions is \(15\) and the product of all integral solutions is \(144\).

### Conclusion
From the options provided:
- (A) sum of all integral solutions is 11 (Incorrect)
- (B) sum of all integral solutions is 16 (Incorrect)
- (C) product of all integral solutions is 144 (Correct)
- (D) product of all integral solutions is 30 (Incorrect)

Thus, the correct answer is (C) product of all integral solutions is 144.