$$\frac{675\times675\times675+325\times325\times325}{67.5\times67.5+32.5\times32.5-67.5\times32.5}$$ is equal to:

Let us denote $$a = 675$$ and $$b = 325$$ for compactness.

Step 1 (Numerator)
The numerator is $$675\times675\times675 + 325\times325\times325 = a^3 + b^3.$$
Use the standard identity $$a^3 + b^3 = (a + b)\,(a^2 - ab + b^2).$$

Therefore, the numerator becomes$$(a + b)\,(a^2 - ab + b^2).$$

Step 2 (Denominator)
Write every term in the denominator in terms of $$a$$ and $$b$$: $$67.5 = \frac{a}{10}, \quad 32.5 = \frac{b}{10}.$$

Hence the denominator is $$\left(\frac{a}{10}\right)^2 + \left(\frac{b}{10}\right)^2 - \left(\frac{a}{10}\right)\left(\frac{b}{10}\right) = \frac{1}{100}\,\bigl(a^2 + b^2 - ab\bigr).$$

Notice that the symmetric expression $$a^2 + b^2 - ab$$ is the same as $$a^2 - ab + b^2,$$ so we can write the denominator as $$\frac{1}{100}\,(a^2 - ab + b^2).$$

Step 3 (Form the quotient)
The entire expression now reads $$\frac{(a + b)\,(a^2 - ab + b^2)}{\dfrac{1}{100}\,(a^2 - ab + b^2)}.$$ The common factor $$a^2 - ab + b^2$$ cancels out, giving $$\frac{a + b}{\dfrac{1}{100}} = 100\,(a + b).$$

Step 4 (Substitute $$a$$ and $$b$$)
$$a + b = 675 + 325 = 1000.$$
Therefore the value of the original expression is $$100 \times 1000 = 100\,000.$$

Hence, the required value is 1,00,000.

Option C which is: 1,00,000