The least value of n for which the number of integral terms in the Binomial expansion of…
The least value of n for which the number of integral terms in the Binomial expansion of \((\sqrt[3]{7}+\sqrt[12]{11})^{\mathrm{n}}\) is 183, is :
2184
2196
2148
2172
Solution
$\begin{aligned}
& \text { General term }={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}\left\{7^{1 / 3}\right\}^{\mathrm{n}-\mathrm{r}}\left(11^{1 / 12}\right)^{\mathrm{r}} \\
& ={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}\{7\}^{\frac{\mathrm{n}-\mathrm{r}}{3}}(11)^{\mathrm{r} / 12}
\end{aligned}$
For integral terms, r must be multiple of 12
$\therefore \mathrm{r}=12 \mathrm{k}, \mathrm{k} \in \mathrm{~W}$
Total values of $r=183$
Hence $\max r=12(182)$
$=2184$
Min value of $n=2184$