Mathematics - Binomial Theorem Question with Solution | TestHub

The general term of multinomial expansion is $\frac{6!}{\alpha !\beta !\gamma !}(1{)}^{\alpha }{\left(2^{\frac{1}{3}}\right)}^{\beta }{\left(3^{\frac{1}{2}}\right)}^{\gamma }$ For terms to be rational $3\mid \beta$ and $2\mid \gamma$ $\begin{array}{cccc}\beta & \gamma & \alpha & \text{ Term }\\ 0& 0& 6& 1\\ 0& 2& 4& 15.3=45\\ 0& 4& 2& 15.3^2=135\\ 0& 6& 0& 1.3^3=27\\ 3& 0& 3& 20.2=40\\ 3& 2& 1& \mathrm{60.2.3}=360\\ 6& 0& 0& 1.4=4\end{array}$ $\Rightarrow$ Sum of rational terms $=1+45+135+27+40+360+4=612$