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 proceeding through a free radical mechanism from the given options, we will analyze each reaction step by step.

### Step-by-Step Solution:

1. **Reaction (i): `C2H5Br + Na+ OCH3-`**
- This is a nucleophilic substitution reaction (SN2) where the methoxide ion (OCH3-) acts as a nucleophile.
- **Mechanism:** Nucleophilic substitution (not free radical).
- **Conclusion:** Not a free radical reaction.

2. **Reaction (ii): `C2H5Br + AgCN`**
- This is also a nucleophilic substitution reaction where AgCN acts as a nucleophile.
- **Mechanism:** Nucleophilic substitution (not free radical).
- **Conclusion:** Not a free radical reaction.

3. **Reaction (iii): `C6H5CH3 + Cl2 (hv)`**
- This reaction involves benzene (C6H5) and chlorine (Cl2) in the presence of light (hv), which promotes free radical formation.
- **Mechanism:** Free radical halogenation.
- **Conclusion:** This is a free radical reaction.

4. **Reaction (iv): `CH3CH=CH2 + HBr (peroxides)`**
- The presence of peroxides leads to the peroxide effect, which results in a free radical mechanism for the addition of HBr to alkenes.
- **Mechanism:** Free radical addition.
- **Conclusion:** This is a free radical reaction.

5. **Reaction (v): `C2H5Br + alc. KOH`**
- This is an elimination reaction (E2) where alcoholic KOH removes a hydrogen halide.
- **Mechanism:** Elimination (not free radical).
- **Conclusion:** Not a free radical reaction.

6. **Reaction (vi): `CH3CH=CH2 + HCl (peroxide)`**
- The peroxide effect does not apply to HCl; hence, this reaction does not proceed via a free radical mechanism.
- **Mechanism:** Not free radical.
- **Conclusion:** Not a free radical reaction.

7. **Reaction (vii): `C2H5Br + H2 (Ni or Pd)`**
- This is a reduction reaction where hydrogen is added to the alkyl bromide.
- **Mechanism:** Reduction (not free radical).
- **Conclusion:** Not a free radical reaction.

8. **Reaction (viii): `C2H6 + Cl2 (hv)`**
- This reaction involves ethane and chlorine in the presence of light, leading to free radical formation and substitution.
- **Mechanism:** Free radical halogenation.
- **Conclusion:** This is a free radical reaction.

9. **Reaction (ix): `C2H5Br + NaSH`**
- This is a nucleophilic substitution reaction where NaSH acts as a nucleophile.
- **Mechanism:** Nucleophilic substitution (not free radical).
- **Conclusion:** Not a free radical reaction.

### Summary of Free Radical Reactions:
- The reactions that proceed through a free radical mechanism are:
- (iii) `C6H5CH3 + Cl2 (hv)`
- (iv) `CH3CH=CH2 + HBr (peroxides)`
- (viii) `C2H6 + Cl2 (hv)`

### Total Count:
- **Total Free Radical Reactions:** 3

### Final Answer:
The number of reactions proceeding through free radical mechanism is **3**.