Statement -1: The expression `n!(20-n)!` is minimum when `n=10`.
because
Statement -2: `.^(2p)C_(r)` is maximum when `r=p`.

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, Statemetn -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

Answer

To solve the problem, we need to analyze both statements and determine their validity step by step.

### Step 1: Analyze Statement 1
**Statement 1**: The expression \( n!(20-n)! \) is minimum when \( n=10 \).

1. **Understanding the Expression**:
The expression \( n!(20-n)! \) represents the product of the factorial of \( n \) and the factorial of \( 20-n \).

2. **Using Combinatorial Identity**:
We can relate this expression to combinations:
\[ n!(20-n)! = \frac{20!}{20Cn} \]
where \( 20Cn \) is the number of ways to choose \( n \) items from 20.

3. **Finding Minimum**:
The expression \( n!(20-n)! \) will be minimized when \( 20Cn \) is maximized. From combinatorial principles, \( 20Cn \) is maximized when \( n \) is around \( \frac{20}{2} = 10 \).

4. **Conclusion for Statement 1**:
Therefore, Statement 1 is true because \( n!(20-n)! \) is indeed minimum when \( n=10 \).

### Step 2: Analyze Statement 2
**Statement 2**: \( 2pC_r \) is maximum when \( r=p \).

1. **Understanding the Expression**:
The expression \( 2pC_r \) represents the number of ways to choose \( r \) items from \( 2p \).

2. **Finding Maximum**:
The binomial coefficient \( 2pC_r \) reaches its maximum value when \( r \) is equal to \( p \) (or close to \( p \)), as this is a property of binomial coefficients.

3. **Conclusion for Statement 2**:
Thus, Statement 2 is also true because \( 2pC_r \) is maximized when \( r=p \).

### Step 3: Relationship Between Statements
- Since both statements are true, we need to check if Statement 2 provides a correct explanation for Statement 1.
- We see that both statements relate to maximizing and minimizing expressions involving factorials and combinations.

### Final Conclusion
Both statements are true, and Statement 2 correctly explains why Statement 1 is true. Therefore, the correct option is that both statements are true, and Statement 2 is a correct explanation for Statement 1.