How many of the following hydroxides are soluble in excess `NaOH` solution?
`Fe(OH)_(3), Al(OH)_(3),Zn(OH)_(3),Zn(OH)_(2), Ni(OH)_(2),Mn(OH)_(2), Sn(OH)_(2), Sn(OH)_(2), Cu(OH)_(2)`

Explanation

To determine how many of the given hydroxides are soluble in excess NaOH solution, we will analyze each hydroxide one by one.

The hydroxides given are:
1. Fe(OH)₃
2. Al(OH)₃
3. Zn(OH)₃
4. Zn(OH)₂
5. Ni(OH)₂
6. Mn(OH)₂
7. Sn(OH)₂
8. Cu(OH)₂

### Step-by-step solution:

1. **Fe(OH)₃ (Iron(III) hydroxide)**:
- Iron(III) hydroxide is generally insoluble in excess NaOH.
- **Conclusion**: Not soluble.

2. **Al(OH)₃ (Aluminum hydroxide)**:
- Aluminum hydroxide is amphoteric and reacts with excess NaOH to form sodium aluminate (NaAlO₂), which is soluble.
- **Conclusion**: Soluble.

3. **Zn(OH)₃ (Zinc(III) hydroxide)**:
- Zinc(III) hydroxide is not commonly referenced; however, zinc hydroxide (Zn(OH)₂) is amphoteric and can dissolve in excess NaOH to form sodium zincate (Na₂ZnO₂).
- **Conclusion**: Not applicable as Zn(OH)₃ is not a common species; we will consider Zn(OH)₂ instead.

4. **Zn(OH)₂ (Zinc(II) hydroxide)**:
- Zinc(II) hydroxide is amphoteric and reacts with excess NaOH to form sodium zincate (Na₂ZnO₂), which is soluble.
- **Conclusion**: Soluble.

5. **Ni(OH)₂ (Nickel(II) hydroxide)**:
- Nickel(II) hydroxide is generally insoluble in excess NaOH.
- **Conclusion**: Not soluble.

6. **Mn(OH)₂ (Manganese(II) hydroxide)**:
- Manganese(II) hydroxide is generally insoluble in excess NaOH.
- **Conclusion**: Not soluble.

7. **Sn(OH)₂ (Tin(II) hydroxide)**:
- Tin(II) hydroxide is amphoteric and reacts with excess NaOH to form sodium stannate (Na₂SnO₂), which is soluble.
- **Conclusion**: Soluble.

8. **Cu(OH)₂ (Copper(II) hydroxide)**:
- Copper(II) hydroxide is generally insoluble in excess NaOH.
- **Conclusion**: Not soluble.

### Summary of Solubility:
- Soluble in excess NaOH: Al(OH)₃, Zn(OH)₂, Sn(OH)₂
- Not soluble in excess NaOH: Fe(OH)₃, Zn(OH)₃ (not applicable), Ni(OH)₂, Mn(OH)₂, Cu(OH)₂

### Final Count:
Out of the given hydroxides, **3 hydroxides (Al(OH)₃, Zn(OH)₂, Sn(OH)₂)** are soluble in excess NaOH.