What is the correct order of decreasing stability of the following cations
I. `CH_(3) - overset(oplus)(C) H - CH_(3)` II. `CH_(3) - overset(oplus)(C)H - OCH_(3)`
III. `CH_(3) - overset(oplus)(C)H - CH_(2) - OCH_(3)`

Options

A. `II gt I gt III`
B. `II gt III gt I`
C. `III gt I gt II`
D. `I gt II gt III`

Explanation

To determine the correct order of decreasing stability of the given cations, we will analyze the structure and the effects of the substituents attached to the cationic carbon.

### Step-by-Step Solution:

1. **Identify the Structures of the Cations**:
- **Cation I**: `CH3 - C^+ - CH3`
- **Cation II**: `CH3 - C^+ - OCH3`
- **Cation III**: `CH3 - C^+ - CH2 - OCH3`

2. **Determine the Degree of Each Cation**:
- All three cations are secondary (2°) carbocations, as each has two carbon atoms attached to the positively charged carbon.

3. **Analyze the Effects of the Substituents**:
- **Cation I**: The methyl groups (`CH3`) exhibit a +I (inductive) effect, which is electron-donating but relatively weak.
- **Cation II**: The methoxy group (`OCH3`) has both +R (resonance) and +I (inductive) effects. The +R effect is stronger than +I, and since the positive charge is adjacent to the methoxy group, it stabilizes the cation significantly through resonance.
- **Cation III**: The methoxy group is further away from the positively charged carbon, and while it can still exhibit +R, the presence of the `CH2` group between them reduces the resonance stabilization. Additionally, the methoxy group can also exhibit a -I effect due to the electronegativity of oxygen, which can pull electron density away from the cation, further destabilizing it.

4. **Compare the Stability**:
- **Cation II** is the most stable due to the strong resonance effect from the methoxy group.
- **Cation I** is next, as it has two methyl groups providing +I effect but lacks the resonance stabilization present in Cation II.
- **Cation III** is the least stable because the methoxy group is less effective at stabilizing the positive charge due to the distance and potential -I effect.

5. **Order of Stability**:
- The order of decreasing stability is: **Cation II > Cation I > Cation III**.
- In numerical terms, this can be represented as: **II > I > III** or **213**.

### Final Answer:
The correct order of decreasing stability of the cations is **Cation II > Cation I > Cation III** (213).