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 $ (1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + ... + a_{20} x^{20} $. If $ (a_1 + a_3 + a_5 + ... + a_{19}) - 11a_2 = 121k $, then k is equal to _______
In the expansion of $\left( \sqrt{5} + \frac{1}{\sqrt{5}} \right)^n$, $n \in \mathbb{N}$, if the ratio of $15^{th}$ term from the beginning to the $15^{th}$ term from the end is $\frac{1}{6}$, then the value of $^nC_3$ is:
Match List-I with List-II: List-I