Question

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 :

(1) 2184

(2) 2148

(3) 2172

(4) 2196

Answer 1

Correct option is (1) 2184  

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

\(\therefore \mathrm{r}=12 \mathrm{k}, \mathrm{k} \in \mathrm{W}\)

Total values of \(\mathrm{r}=183\)

Hence \(\max r=12(182)\) 

=2184