Which one of the following is correct ?

Options

A. a. `25^(26) gt 26^(25)`
B. b. `100^(101) lt 101^(100)`
C. c. `199^(198) gt 198^(199)`
D. d. `399^(400) lt 400^(399)`

Answer

To solve the problem, we need to analyze each option and determine which statement is correct based on the relationship between the bases and their respective exponents.

### Step-by-Step Solution:

1. **Analyze Option A: \(25^{26} \) vs \(26^{25}\)**
- Here, we compare \(25\) and \(26\). Since \(25 < 26\), we need to check the powers.
- According to the principle, if a smaller base is raised to a higher power, it can be greater than a larger base raised to a lower power.
- Therefore, we compare \(25^{26}\) and \(26^{25}\). Since \(25 < 26\) and \(26\) is raised to a lower power, \(25^{26}\) is indeed greater than \(26^{25}\).
- **Conclusion:** Option A is correct.

2. **Analyze Option B: \(100^{101} \) vs \(101^{100}\)**
- Here, we compare \(100\) and \(101\). Since \(100 < 101\), we again check the powers.
- According to the principle, \(100^{101}\) should be greater than \(101^{100}\) because \(100\) is raised to a higher power.
- However, the statement claims \(100^{101} < 101^{100}\), which is incorrect.
- **Conclusion:** Option B is incorrect.

3. **Analyze Option C: \(199^{198} \) vs \(198^{199}\)**
- Here, we compare \(199\) and \(198\). Since \(199 > 198\), we check the powers.
- According to the principle, \(199^{198}\) should be less than \(198^{199}\) because \(198\) is raised to a higher power.
- The statement claims \(199^{198} > 198^{199}\), which is incorrect.
- **Conclusion:** Option C is incorrect.

4. **Analyze Option D: \(399^{400} \) vs \(400^{399}\)**
- Here, we compare \(399\) and \(400\). Since \(399 < 400\), we check the powers.
- According to the principle, \(399^{400}\) should be less than \(400^{399}\) because \(400\) is raised to a higher power.
- The statement claims \(399^{400} < 400^{399}\), which is correct.
- **Conclusion:** Option D is correct.

### Final Answer:
The correct answer is **Option A: \(25^{26} > 26^{25}\)**.