The least value of n for which the number of integral terms in the Binomial expansion of (…

Question

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 :

Accepted Answer

General term: $\binom{n}{r}(7^{1/3})^{n-r}(11^{1/12})^r=\binom{n}{r}7^{(n-r)/3}11^{r/12}$.

For an integral term, we need $\dfrac{n-r}{3}$ and $\dfrac{r}{12}$ both to be integers. Since $r$ must be a multiple of 12 (say $r=12k$), and we want exactly 183 such terms ($k=0,1,\dots,182$), the maximum $r$ is $12	imes182=2184$, requiring $n\ge2184$.

The least such $n$ is $\boxed{2184}$.

Solutions · 1