The decreasing order of the rate of bromination of the following compounds is:
I. `PhN^(o+)Me_(3)`, II. `PhCH_(2) NMe_(3)`
III. `PhMe`, IV. `PhNMe_(2)`

Options

A. `(I) gt (II) gt (III) gt (IV)`
B. `(IV) gt (III) gt (II) gt (I)`
C. `(III) gt (IV) gt (I) gt (II)`
D. `(III) gt (IV) gt (II) gt (I)`

Explanation

To determine the decreasing order of the rate of bromination of the given compounds, we need to analyze the electronic effects of the substituents on the benzene ring in each compound. Bromination is an electrophilic substitution reaction that is influenced by the electron density on the aromatic ring. Electron-donating groups increase the rate of bromination, while electron-withdrawing groups decrease it.

### Step-by-Step Solution:

1. **Identify the Compounds:**
- I. `PhN^(o+)Me_(3)` (Ph = phenyl, N^(o+) = positively charged nitrogen with three methyl groups)
- II. `PhCH_(2)NMe_(3)` (Ph = phenyl, CH2 = benzylic carbon, NMe3 = nitrogen with three methyl groups)
- III. `PhMe` (Ph = phenyl, Me = methyl group)
- IV. `PhNMe_(2)` (Ph = phenyl, NMe2 = nitrogen with two methyl groups)

2. **Analyze Compound I: `PhN^(o+)Me_(3)`**
- The nitrogen is positively charged and has no lone pairs to donate. It acts as an electron-withdrawing group, decreasing the electron density on the benzene ring.
- **Effect on Bromination:** Decreases the rate.

3. **Analyze Compound II: `PhCH_(2)NMe_(3)`**
- The nitrogen is also positively charged and is attached to a benzylic carbon. Similar to compound I, it acts as an electron-withdrawing group.
- **Effect on Bromination:** Decreases the rate, but slightly less than compound I due to the benzylic position.

4. **Analyze Compound III: `PhMe`**
- The methyl group (Me) is an electron-donating group. It increases the electron density on the benzene ring.
- **Effect on Bromination:** Increases the rate.

5. **Analyze Compound IV: `PhNMe_(2)`**
- The nitrogen has two methyl groups and retains a lone pair, making it a strong electron-donating group.
- **Effect on Bromination:** Significantly increases the rate.

6. **Order the Compounds Based on Their Effects:**
- Compound IV (`PhNMe_(2)`) has the highest rate due to strong electron donation.
- Compound III (`PhMe`) has a moderate rate due to the electron-donating effect of the methyl group.
- Compound II (`PhCH_(2)NMe_(3)`) has a lower rate than III but higher than I due to the benzylic position.
- Compound I (`PhN^(o+)Me_(3)`) has the lowest rate due to the strong electron-withdrawing effect of the positively charged nitrogen.

### Final Order:
The decreasing order of the rate of bromination is:
**IV > III > II > I**

### Answer:
**Option B: IV > III > II > I**

---