Option 3 : \((\frac{27p^3}{8}) - (\frac{125q^3}{27}) -(\frac{25p^2q}{2}) +(\frac{25pq^2}{2})\)

**Formula Used:**

(x - y)^{3} = x^{3} - y^{3} - 3x^{2}y + 3xy^{2}

**Calculation:**

\((\frac{3p}{2} - \frac{5q}{3})^3 -(\frac{5p^2q}{4})\)

\(⇒ (\frac{3p}{2})^3 - (\frac{5q}{3})^3 -3(\frac{3p}{2})^2(\frac{5q}{3}) +3(\frac{3p}{2})(\frac{5q}{3})^2 -(\frac{5p^2q}{4})\)

\(⇒ (\frac{27p^3}{8}) - (\frac{125q^3}{27}) -(\frac{45p^2q}{4}) +(\frac{25pq^2}{2}) -(\frac{5p^2q}{4})\)

\(⇒ (\frac{27p^3}{8}) - (\frac{125q^3}{27}) -(\frac{25p^2q}{2}) +(\frac{25pq^2}{2})\)

\(∴ (\frac{3p}{2} - \frac{5q}{3})^3 -(\frac{5p^2q}{4})\) = \((\frac{27p^3}{8}) - (\frac{125q^3}{27}) -(\frac{25p^2q}{2}) +(\frac{25pq^2}{2})\)

The correct option is 3 i.e. \((\frac{27p^3}{8}) - (\frac{125q^3}{27}) -(\frac{25p^2q}{2}) +(\frac{25pq^2}{2})\)