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 the two statements given and determine their validity.

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

To analyze this, we can rewrite the expression in terms of combinations:
\[ \binom{20}{n} = \frac{20!}{n!(20-n)!} \]
Thus, we can express \( n!(20-n)! \) as:
\[ n!(20-n)! = \frac{20!}{\binom{20}{n}} \]
This means that \( n!(20-n)! \) is minimized when \( \binom{20}{n} \) is maximized.

### Step 2: Find the Maximum of \( \binom{20}{n} \)
The binomial coefficient \( \binom{20}{n} \) reaches its maximum when \( n \) is around \( \frac{20}{2} = 10 \). This is a well-known property of binomial coefficients, where they are symmetric and peak at the middle values.

### Step 3: Conclusion for Statement 1
Since \( \binom{20}{n} \) is maximized at \( n = 10 \), it follows that \( n!(20-n)! \) is minimized at \( n = 10 \). Therefore, **Statement 1 is true**.

### Step 4: Analyze Statement 2
**Statement 2:** \( \binom{2p}{r} \) is maximum when \( r=p \).

This is also a known property of binomial coefficients. The binomial coefficient \( \binom{n}{k} \) is maximized when \( k \) is approximately \( \frac{n}{2} \). Therefore, for \( n = 2p \), it is maximized when \( r = p \).

### Step 5: Conclusion for Statement 2
Since \( \binom{2p}{r} \) is indeed maximized at \( r = p \), **Statement 2 is true**.

### Final Conclusion
Both statements are true. However, Statement 2 does not serve as a correct explanation for Statement 1, as they pertain to different contexts. Thus, the answer is:

**Option A:** Statement 1 is true, Statement 2 is true, but Statement 2 is not a correct explanation of Statement 1.

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