The problem asks to express the fraction $\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers.

1. Problem Description

The problem asks to express the fraction $\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers.

2. Solution Steps

To express the given fraction in the required form, we need to rationalize the denominator. We multiply both the numerator and the denominator by the conjugate of the denominator, which is $2\sqrt{3} + \sqrt{2}$.

$\frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}} = \frac{3\sqrt{2} - \sqrt{3}}{2\sqrt{3} - \sqrt{2}} \cdot \frac{2\sqrt{3} + \sqrt{2}}{2\sqrt{3} + \sqrt{2}}$

Now, multiply the numerators and denominators:

Numerator: $(3\sqrt{2} - \sqrt{3})(2\sqrt{3} + \sqrt{2}) = 3\sqrt{2} \cdot 2\sqrt{3} + 3\sqrt{2} \cdot \sqrt{2} - \sqrt{3} \cdot 2\sqrt{3} - \sqrt{3} \cdot \sqrt{2} = 6\sqrt{6} + 6 - 6 - \sqrt{6} = 5\sqrt{6}$

Denominator: $(2\sqrt{3} - \sqrt{2})(2\sqrt{3} + \sqrt{2}) = (2\sqrt{3})^2 - (\sqrt{2})^2 = 4 \cdot 3 - 2 = 12 - 2 = 10$

So, we have:

$\frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2}$

Now, we want to express this in the form $\frac{\sqrt{m}}{\sqrt{n}}$. We can write $2$ as $\sqrt{4}$.

$\frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}}$

If we multiply both the numerator and denominator by $\sqrt{2.5}$ we get:

$\frac{\sqrt{6}}{\sqrt{4}} = \frac{\sqrt{6} \cdot \sqrt{2.5}}{\sqrt{4} \cdot \sqrt{2.5}} = \frac{\sqrt{15}}{\sqrt{10}}$.

This is not one of the choices

The options are:

(a) $\frac{\sqrt{6}}{\sqrt{10}}$

(b) $\frac{\sqrt{150}}{\sqrt{100}}$

(c) $\frac{2\sqrt{6}}{\sqrt{10}}$

(d) $\frac{5\sqrt{6}}{\sqrt{10}}$

Let's simplify each option to see which one is equal to $\frac{\sqrt{6}}{2}$.

(a) $\frac{\sqrt{6}}{\sqrt{10}} = \frac{\sqrt{6}}{\sqrt{10}}$

(b) $\frac{\sqrt{150}}{\sqrt{100}} = \frac{\sqrt{25 \cdot 6}}{10} = \frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2}$

(c) $\frac{2\sqrt{6}}{\sqrt{10}}$

(d) $\frac{5\sqrt{6}}{\sqrt{10}}$

3. Final Answer

The correct answer is (b) $\frac{\sqrt{150}}{\sqrt{100}}$.