Consider the following two statements :
I. if n is a composite number, then n divides (n – 1)!
II. There are infinitely many natural numbers n such that `n^3 + 2n^2 + n` divides n!.
Then

Options

A. I and II are true
B. I and II are false
C. I is true and II is false
D. I is false and II is true

Answer

To solve the problem, we need to evaluate the two statements provided and determine their validity.

### Step 1: Evaluate Statement I
**Statement I:** If \( n \) is a composite number, then \( n \) divides \( (n - 1)! \).

1. **Understanding Factorials**: The factorial \( (n - 1)! \) is the product of all positive integers from 1 to \( n - 1 \).
2. **Composite Number Definition**: A composite number \( n \) has factors other than 1 and itself, meaning it can be expressed as a product of smaller integers.
3. **Example**: Let's take \( n = 6 \) (a composite number):
- Calculate \( (n - 1)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
- Check if \( 6 \) divides \( 120 \): \( 120 \div 6 = 20 \), which is an integer.
4. **Conclusion for Statement I**: Since this holds true for the example and can be generalized for any composite \( n \) (as it will always have factors in \( (n - 1)! \)), **Statement I is true**.

### Step 2: Evaluate Statement II
**Statement II:** There are infinitely many natural numbers \( n \) such that \( n^3 + 2n^2 + n \) divides \( n! \).

1. **Rewriting the Expression**: The expression \( n^3 + 2n^2 + n \) can be factored:
\[ n^3 + 2n^2 + n = n(n^2 + 2n + 1) = n(n + 1)^2 \]
2. **Divisibility Condition**: We need to check if \( n! \) is divisible by \( n(n + 1)^2 \).
3. **Choosing Values for \( n \)**: Let's take \( n = 11 \):
- Calculate \( n! = 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).
- Calculate \( n(n + 1)^2 = 11 \times 12^2 = 11 \times 144 = 1584 \).
4. **Check Divisibility**: We need to see if \( 11! \) is divisible by \( 1584 \):
- Simplifying, \( 11! \) contains the factors \( 11, 12, 12 \) (from \( 12^2 \)), thus it is divisible by \( 1584 \).
5. **Conclusion for Statement II**: Since we can find such \( n \) (like \( n = 11 \)) and there are infinitely many natural numbers satisfying this condition, **Statement II is true**.

### Final Conclusion
Both statements are true. Therefore, the answer is that both statements are correct.