Statement-1: The smallest positive integer n such that n! can be expressed as a product of n-3 consecutive integers, is 6.
Statement-2: Product of three consecutive integers is divisible by 6.

Options

A. Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1
B. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1
C. Statement-1 is true, statement-2 is false
D. Statement-1 is false, statement-2 is true

Explanation

To solve the problem, we need to analyze both statements provided in the question.

### Step 1: Analyze Statement-1
The first statement claims that the smallest positive integer \( n \) such that \( n! \) can be expressed as a product of \( n - 3 \) consecutive integers is 6.

1. **Calculate \( 6! \)**:
\[ 6! = 720 \]

2. **Determine \( n - 3 \)**:
\[ n - 3 = 6 - 3 = 3 \]
So, we need to express \( 720 \) as a product of 3 consecutive integers.

3. **Find 3 consecutive integers whose product is 720**:
Let the three consecutive integers be \( x, x + 1, x + 2 \).
\[ x(x + 1)(x + 2) = 720 \]
Testing values, we find:
\[ 8 \times 9 \times 10 = 720 \]
Thus, \( 720 \) can indeed be expressed as the product of 3 consecutive integers.

4. **Check smaller values of \( n \)**:
- For \( n = 5 \):
\[ 5! = 120 \quad \text{and} \quad n - 3 = 2 \]
We need to express \( 120 \) as a product of 2 consecutive integers:
\[ x(x + 1) = 120 \]
This leads to the equation:
\[ x^2 + x - 120 = 0 \]
Using the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{1 + 480}}{2} = \frac{-1 \pm 22}{2} \]
The solutions are \( x = 10.5 \) (not an integer) and \( x = -11.5 \) (not valid). Thus, \( 120 \) cannot be expressed as a product of 2 consecutive integers.

Since \( n = 6 \) is the smallest integer for which \( n! \) can be expressed as a product of \( n - 3 \) consecutive integers, **Statement-1 is true**.

### Step 2: Analyze Statement-2
The second statement claims that the product of 3 consecutive integers is divisible by 6.

1. **Understanding the product of 3 consecutive integers**:
Let the three consecutive integers be \( x, x + 1, x + 2 \).
The product is:
\[ x(x + 1)(x + 2) \]

2. **Divisibility by 6**:
- Among any three consecutive integers, at least one of them is even, ensuring divisibility by 2.
- Additionally, at least one of the integers is divisible by 3.

Since the product of 3 consecutive integers is divisible by both 2 and 3, it follows that it is divisible by:
\[ 2 \times 3 = 6 \]
Thus, **Statement-2 is also true**.

### Conclusion
Both statements are true, but they are not directly related to each other. Therefore, the answer is that both statements are true, and Statement-2 does not explain Statement-1.