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JEE Main 202529 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
29 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2025 · 29 January · Morning Shift · Mathematics · Binomial Theorem

The least value of n for which the number of integral terms in the...

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 :

Options

  1. A2184
  2. B2196
  3. C2148
  4. D2172

Correct answer

A. 2184

Step-by-step solution

 General term =nCr{71/3}n−r(111/12)r=nCr{7}n−r3(11)r/12\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 ∴r=12k,k∈WTotalvaluesof\therefore \mathrm{r}=12 \mathrm{k}, \mathrm{k} \in \mathrm{ W} Total values ofr=183HenceHence\max r=12(182) =2184MinvalueofMin value ofn=2184
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