The correct order of increasing basicity of the following is
`H_(2)O, OH^(-), CH_(3)OH, CH_(3)O^(-)`

Options

A. `CH_(3)OH lt H_(2)O lt OH^(-) lt CH_(3)O^(-)`
B. `H_(2)O lt CH_(3)OH lt CH_(3)O^(-) lt OH^(-)`
C. `H_(2)O lt CH_(3)OH lt OH^(-) lt CH_(3)O^(-)`
D. `CH_(3)OH lt H_(2)O lt CH_(3)O^(-) lt OH^(-)`

Explanation

To determine the correct order of increasing basicity among the given compounds: H₂O, OH⁻, CH₃OH, and CH₃O⁻, we need to analyze the basicity of each compound based on their ability to donate a lone pair of electrons and accept protons (H⁺).

### Step-by-Step Solution:

1. **Understanding Basicity**: Basicity refers to the ability of a species to accept protons (H⁺) or donate electron pairs. A stronger base is more likely to donate its electrons or accept protons.

2. **Analyzing CH₃O⁻ (Methoxide Ion)**:
- CH₃O⁻ has a negative charge on oxygen, which increases its electron density, making it very reactive and a strong base.
- The presence of the methyl group (CH₃) enhances the electron density on oxygen due to the +I (inductive) effect, making it more likely to donate its electrons.

3. **Analyzing OH⁻ (Hydroxide Ion)**:
- OH⁻ also has a negative charge on oxygen, but it is less basic than CH₃O⁻.
- The hydroxide ion is a strong base, but the electron-donating effect of the hydrogen atom is weaker compared to the methyl group in CH₃O⁻.

4. **Analyzing CH₃OH (Methanol)**:
- CH₃OH has a hydroxyl group (-OH) but lacks a negative charge.
- While it can donate a lone pair, it is less basic than both OH⁻ and CH₃O⁻ because it does not have a negative charge to facilitate proton acceptance.

5. **Analyzing H₂O (Water)**:
- H₂O is the least basic among the four.
- It can donate a lone pair but does not have a negative charge, making it less likely to accept protons compared to the other species.

6. **Final Order of Basicity**:
- Based on the analysis, we can arrange the compounds in order of increasing basicity:
- H₂O < CH₃OH < OH⁻ < CH₃O⁻

### Conclusion:
The correct order of increasing basicity is:
**H₂O < CH₃OH < OH⁻ < CH₃O⁻**