The least value of $n$ for which the number of integral terms in the Binomial expansion of $\left(\sqrt[3]{7} + \sqrt[12]{11}\right)^n$ is 183, is:
Step-by-Step Solution
Key Concept: For the general term ${}^nC_r \cdot 7^{(n-r)/3} \cdot 11^{r/12}$ to be integral, $r$ must be a multiple of LCM(3,12) = 12.
General term $= {}^nC_r \cdot 7^{(n-r)/3} \cdot 11^{r/12}$. For integral terms, $r = 12k$, $k \in \mathbb{W}$. Total integral terms $= \lfloor n/12 \rfloor + 1 = 183 \Rightarrow n/12 = 182 \Rightarrow$ min $n = 2184$.
Correct Answer: 2184