What is the principal?
I. The simple interest accrued on that sum at the rare of `12%` per annum in 2 years is Rs. 360 less than the compoind interest on the same sum at `12%` per annum in 2 years.
II. The sum doubles itself in 10 years at `10%` per annum rate of simple interest.
III. The compoind interest on teh sum in 2 years at the rate of `12%` per annum is Rs.6360.

Options

A. I or II
B. I and III
C. I and II
D. II and III

Explanation

To find the principal amount based on the given statements, we will analyze each statement one by one.

### Step 1: Analyze Statement I
Statement I states that the simple interest accrued on the sum at the rate of 12% per annum in 2 years is Rs. 360 less than the compound interest on the same sum at 12% per annum in 2 years.

1. **Calculate Simple Interest (SI)**:
- Formula: \( SI = \frac{P \times R \times T}{100} \)
- Here, \( R = 12\% \) and \( T = 2 \) years.
- Therefore, \( SI = \frac{P \times 12 \times 2}{100} = \frac{24P}{100} = 0.24P \).

2. **Calculate Compound Interest (CI)**:
- Formula: \( CI = P \left(1 + \frac{R}{100}\right)^T - P \)
- Here, \( R = 12\% \) and \( T = 2 \) years.
- Therefore, \( CI = P \left(1 + \frac{12}{100}\right)^2 - P = P \left(1.12^2 - 1\right) = P(1.2544 - 1) = P \times 0.2544 \).

3. **Set up the equation**:
- According to the statement: \( SI = CI - 360 \)
- Thus, \( 0.24P = 0.2544P - 360 \).

4. **Solve for P**:
- Rearranging gives: \( 360 = 0.2544P - 0.24P \)
- This simplifies to: \( 360 = 0.0144P \)
- Therefore, \( P = \frac{360}{0.0144} = 25000 \).

### Step 2: Analyze Statement II
Statement II states that the sum doubles itself in 10 years at 10% per annum rate of simple interest.

1. **Using the formula for Simple Interest**:
- If the sum doubles, then the total amount \( A = 2P \).
- The formula for Simple Interest is \( A = P + SI \).
- Therefore, \( 2P = P + SI \) implies \( SI = P \).

2. **Calculate SI**:
- Using the formula \( SI = \frac{P \times R \times T}{100} \):
- Here, \( R = 10\% \) and \( T = 10 \) years.
- Thus, \( SI = \frac{P \times 10 \times 10}{100} = P \).

3. **Set up the equation**:
- From \( SI = P \), we have \( P = P \), which is always true but does not provide a specific value for P.

### Step 3: Analyze Statement III
Statement III states that the compound interest on the sum in 2 years at the rate of 12% per annum is Rs. 6360.

1. **Using the formula for Compound Interest**:
- We have already derived that \( CI = P \times 0.2544 \).
- Setting this equal to Rs. 6360 gives:
- \( 0.2544P = 6360 \).

2. **Solve for P**:
- Therefore, \( P = \frac{6360}{0.2544} \approx 25000 \).

### Conclusion
From the analysis of the statements:
- Statement I gives us \( P = 25000 \).
- Statement II does not provide a specific value for P.
- Statement III also gives us \( P = 25000 \).

Thus, we can conclude that the principal amount is Rs. 25000 based on Statements I and III.