Quantity I: `165m^(5)n^(3)div55/3m^(-2)n^(5)8m^(-5)n^(8),mgt0,nlt0`
Quantity II: `210x^(9)y^(9)div35x^(4)y^(2)div3x^(2)y^(4)'xlt0,ylt0`
Quantity III: `68a^(8)b^(12)xx5a^(3)b^(-4)div17a^(6)b,agt0,blt0`

Options

A. (@,&)
B. `(#,@)
C. ($,@)
D. (&,@)

Explanation

To solve the problem, we need to evaluate each of the three quantities step by step.

### Quantity I:
\[ \text{Quantity I} = \frac{165 m^5 n^3}{\frac{55}{3} m^{-2} n^5 \cdot 8 m^{-5} n^8} \]

1. **Simplify the denominator**:
- The denominator can be rewritten as:
\[ \frac{55}{3} m^{-2} n^5 \cdot 8 m^{-5} n^8 = \frac{55 \cdot 8}{3} m^{-2} m^{-5} n^{5 + 8} = \frac{440}{3} m^{-7} n^{13} \]

2. **Rewrite the expression**:
- Now we can rewrite Quantity I as:
\[ \frac{165 m^5 n^3}{\frac{440}{3} m^{-7} n^{13}} = 165 m^5 n^3 \cdot \frac{3}{440} m^{7} n^{-13} \]

3. **Combine the terms**:
- This simplifies to:
\[ \frac{165 \cdot 3}{440} m^{5 + 7} n^{3 - 13} = \frac{495}{440} m^{12} n^{-10} \]

4. **Further simplify the fraction**:
- The fraction \(\frac{495}{440}\) can be simplified:
\[ \frac{495}{440} = \frac{99}{88} \quad (\text{dividing by 5}) \]

5. **Final expression for Quantity I**:
- Thus, Quantity I becomes:
\[ \frac{99}{88} m^{12} n^{-10} \]

### Quantity II:
\[ \text{Quantity II} = \frac{210 x^9 y^9}{35 x^4 y^2 \cdot 3 x^2 y^4} \]

1. **Simplify the denominator**:
- The denominator simplifies to:
\[ 35 x^4 y^2 \cdot 3 x^2 y^4 = 105 x^{4 + 2} y^{2 + 4} = 105 x^6 y^6 \]

2. **Rewrite the expression**:
- Now we can rewrite Quantity II as:
\[ \frac{210 x^9 y^9}{105 x^6 y^6} = 2 x^{9 - 6} y^{9 - 6} = 2 x^3 y^3 \]

### Quantity III:
\[ \text{Quantity III} = \frac{68 a^8 b^{12}}{17 a^6 b} \cdot 5 a^3 b^{-4} \]

1. **Simplify the first part**:
- The first part simplifies to:
\[ \frac{68 a^8 b^{12}}{17 a^6 b} = 4 a^{8 - 6} b^{12 - 1} = 4 a^2 b^{11} \]

2. **Combine with the second part**:
- Now we combine with the second part:
\[ 4 a^2 b^{11} \cdot 5 a^3 b^{-4} = 20 a^{2 + 3} b^{11 - 4} = 20 a^5 b^7 \]

### Summary of Quantities:
- **Quantity I**: \(\frac{99}{88} m^{12} n^{-10}\)
- **Quantity II**: \(2 x^3 y^3\)
- **Quantity III**: \(20 a^5 b^7\)