The number of reactions proceeding through free radical mechanism are
(i) `C_(2)H_(5)Br+Na^(+)""^(-)OCH_(3)`
(ii) `C_(2)H_(5)Br+AgCN`
(iii) `C_(6)H_(5)CH_(3)+Cl_(2)(hv)`
(iv) `CH_(3)CH=CH_(2)+HBr` (peroxides)
(v) `C_(2)H_(5)Br+alc.KOH`
(vi) `CH_(3)CH=CH_(2)+HCl`(peroxide)
(vii) `C_(2)H_(5)Br+H_(2)(Ni "or" Pd)`
(viii) `C_(2)H_(6)+Cl_(2)(hv)`
(ix) `C_(2)H_(5)Br+NaSH`

Explanation

To determine the number of reactions that proceed through a free radical mechanism from the given options, we will analyze each reaction based on the conditions that favor free radical formation, such as the presence of light, heat, or peroxides.

### Step-by-Step Solution:

1. **Reaction (i): `C2H5Br + Na+ OCH3-`**
- This reaction involves a nucleophilic substitution where the methoxide ion (OCH3-) attacks the carbon attached to bromine. No free radicals are generated here.
- **Conclusion:** Not a free radical mechanism.

2. **Reaction (ii): `C2H5Br + AgCN`**
- Similar to the first reaction, this involves a nucleophilic attack by AgCN on the carbon attached to bromine. No free radicals are formed.
- **Conclusion:** Not a free radical mechanism.

3. **Reaction (iii): `C6H5CH3 + Cl2 (hv)`**
- The presence of light (hv) indicates that this reaction can proceed through a free radical mechanism. Chlorine radicals are generated, which can abstract hydrogen from toluene, leading to the formation of chlorinated products.
- **Conclusion:** This is a free radical mechanism.

4. **Reaction (iv): `CH3CH=CH2 + HBr (peroxides)`**
- The presence of peroxides suggests that this reaction can proceed via a free radical mechanism, where the peroxide initiates the formation of bromine radicals that add to the alkene.
- **Conclusion:** This is a free radical mechanism.

5. **Reaction (v): `C2H5Br + alc. KOH`**
- Alcoholic KOH typically leads to an elimination reaction (E2) rather than a free radical mechanism. No radicals are generated.
- **Conclusion:** Not a free radical mechanism.

6. **Reaction (vi): `CH3CH=CH2 + HCl (peroxide)`**
- Similar to reaction (iv), the presence of peroxides allows for a free radical mechanism where HCl can dissociate into radicals.
- **Conclusion:** This is a free radical mechanism.

7. **Reaction (vii): `C2H5Br + H2 (Ni or Pd)`**
- This reaction involves hydrogenation, which does not involve free radicals. It proceeds through a different mechanism (catalytic).
- **Conclusion:** Not a free radical mechanism.

8. **Reaction (viii): `C2H6 + Cl2 (hv)`**
- The presence of light indicates that chlorine radicals will be generated, leading to the substitution of hydrogen atoms in ethane.
- **Conclusion:** This is a free radical mechanism.

9. **Reaction (ix): `C2H5Br + NaSH`**
- This is a nucleophilic substitution reaction where the thiolate ion (SH-) attacks the carbon attached to bromine. No free radicals are formed.
- **Conclusion:** Not a free radical mechanism.

### Summary of Reactions with Free Radical Mechanism:
- Reactions that proceed through free radical mechanisms: (iii), (iv), (vi), (viii).
- Total number of reactions proceeding through free radical mechanism: **4**

### Final Answer:
The number of reactions proceeding through free radical mechanism is **4**.