Mathematics · Binomial Theorem

JEE Main 2025 — 29 January, Morning Shift — Question 52

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 :

Answer: A

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}

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 $n=2184$

Answer key and solution verified before publishing.

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