Which one of the following is correct ?
`(i) (123)^(369) > (369)^(123) " " (II) (246)^(642) < (642)^(246)`
`(iii) (50)^(50!) = (50!)^(50) " " (iv) 3^(4^(5)) > 5^(4^(3))`

Options

A. (i) and (ii)
B. (ii) and (iii)
C. (i) only
D. (i) and (iv)

Answer

To determine which of the given statements is correct, we will analyze each statement one by one using logarithmic properties.

### Step-by-step Solution:

**Statement (i):** \( (123)^{369} > (369)^{123} \)

1. Take logarithm on both sides:
\[ \log((123)^{369}) > \log((369)^{123}) \]
2. Using the property of logarithms, we can rewrite this as:
\[ 369 \log(123) > 123 \log(369) \]
3. Calculate \( \log(123) \) and \( \log(369) \):
- \( \log(123) \approx 2.08 \)
- \( \log(369) \approx 2.56 \)
4. Substitute these values into the inequality:
\[ 369 \times 2.08 \quad \text{and} \quad 123 \times 2.56 \]
- \( 369 \times 2.08 \approx 766.32 \)
- \( 123 \times 2.56 \approx 314.88 \)
5. Since \( 766.32 > 314.88 \), the statement is true.

**Statement (ii):** \( (246)^{642} < (642)^{246} \)

1. Take logarithm on both sides:
\[ \log((246)^{642}) < \log((642)^{246}) \]
2. Rewrite using logarithmic properties:
\[ 642 \log(246) < 246 \log(642) \]
3. Calculate \( \log(246) \) and \( \log(642) \):
- \( \log(246) \approx 2.39 \)
- \( \log(642) \approx 2.80 \)
4. Substitute these values into the inequality:
\[ 642 \times 2.39 \quad \text{and} \quad 246 \times 2.80 \]
- \( 642 \times 2.39 \approx 1534.38 \)
- \( 246 \times 2.80 \approx 688.80 \)
5. Since \( 1534.38 > 688.80 \), the statement is false.

**Statement (iii):** \( (50)^{(50!)} = (50!)^{50} \)

1. Take logarithm on both sides:
\[ \log((50)^{(50!)}) = \log((50!)^{50}) \]
2. Rewrite using logarithmic properties:
\[ 50! \log(50) = 50 \log(50!) \]
3. Since \( 50! \) is a very large number, we can conclude that \( 50! \log(50) \) will not equal \( 50 \log(50!) \). Therefore, this statement is false.

**Statement (iv):** \( 3^{(4^5)} > 5^{(4^3)} \)

1. Take logarithm on both sides:
\[ \log(3^{(4^5)}) > \log(5^{(4^3)}) \]
2. Rewrite using logarithmic properties:
\[ 4^5 \log(3) > 4^3 \log(5) \]
3. Calculate \( 4^5 = 1024 \) and \( 4^3 = 64 \).
4. Substitute these values into the inequality:
\[ 1024 \log(3) > 64 \log(5) \]
5. Calculate \( \log(3) \approx 0.477 \) and \( \log(5) \approx 0.699 \):
- \( 1024 \times 0.477 \approx 488.928 \)
- \( 64 \times 0.699 \approx 44.736 \)
6. Since \( 488.928 > 44.736 \), the statement is true.

### Conclusion:
The correct statements are (i) and (iv).