In the binomial expansion of $(a+b)^{18}$, the general term is given by $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.
Using the formula for the general term in the binomial expansion, we can find the seventh and thirteenth terms of the given expansion.
Seventh term:
$T_7 = \binom{18}{6} \left(\frac{1}{3x^{\frac{1}{3}}}\right)^{12} \left(\frac{1}{2x^{\frac{1}{3}}}\right)^6$
$m = \binom{18}{6} \left(\frac{1}{3}\right)^{12} \left(\frac{1}{2}\right)^6$
Thirteenth term:
$T_{13} = \binom{18}{12} \left(\frac{1}{3x^{\frac{1}{3}}}\right)^6 \left(\frac{1}{2x^{\frac{1}{3}}}\right)^{12}$
$n = \binom{18}{12} \left(\frac{1}{3}\right)^6 \left(\frac{1}{2}\right)^{12}$
Now, we need to find $\left(\frac{n}{m}\right)^{\frac{1}{3}}$.
$\left(\frac{n}{m}\right)^{\frac{1}{3}} = \left(\frac{\binom{18}{12} \left(\frac{1}{3}\right)^6 \left(\frac{1}{2}\right)^{12}}{\binom{18}{6} \left(\frac{1}{3}\right)^{12} \left(\frac{1}{2}\right)^6}\right)^{\frac{1}{3}}$
Simplifying the expression, we get:
$\left(\frac{n}{m}\right)^{\frac{1}{3}} = \left(\frac{\binom{18}{12}}{\binom{18}{6}} \times \left(\frac{1}{3}\right)^{-6} \times \left(\frac{1}{2}\right)^6\right)^{\frac{1}{3}}$
Using the property of binomial coefficients $\binom{n}{r} = \binom{n}{n-r}$, we can simplify further:
$\left(\frac{n}{m}\right)^{\frac{1}{3}} = \left(\frac{\binom{18}{6}}{\binom{18}{6}} \times \left(\frac{1}{3}\right)^{-6} \times \left(\frac{1}{2}\right)^6\right)^{\frac{1}{3}} = \left(\frac{1}{3^{-6}} \times \frac{1}{2^6}\right)^{\frac{1}{3}} = \left(3^6 \times 2^{-6}\right)^{\frac{1}{3}} = \left(\frac{3^2}{2^2}\right) = \frac{9}{4}$
Therefore, the correct answer is (4).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32