Least value of n for which number of integral terms in Binomial expansion of (√[3]{7}+√[12]{11})

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7.Binomial Theorem

hard

The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^{ n }$ is $183$ , is :

A$2184$

B$2148$

C$2172$

D$2196$

(JEE MAIN-2025)

Solution

General term $={ }^n C _{ r }\left\{7^{1 / 3}\right\}^{ n – r }\left(11^{1 / 12}\right)^{ r }$
$={ }^{n} C_{r}\{7\}^{\frac{n-r}{3}}(11)^{r / 12}$
For integral terms, r must be multiple of $12$
$\therefore r=12 k, k \in W$
Total values of $r =183$
Hence $\max r =12(182)$
$=2184$
Min value of $n=2184$

Standard 11

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Solution

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