Mathematics - Binomial Theorem Question with Solution | TestHub

Given expansion is ${\left(y^{1/5}+x^{1/10}\right)}^{55}$ The general term is $$T_{r+1}={}^{55}{\mathrm{C}}_r{\left(y^{1/5}\right)}^{55-r}\cdot {\left(x^{\frac{1}{10}}\right)}^r$$ $T_{r+1}$ would free from radical sign if powers of $y$ and $x$ are integers. i.e. $\frac{55-r}{5}$ and $\frac{r}{10}$ are integer. $\Rightarrow r$ is multiple of 10 . Hence, $r=0,10,20,30,40,50$ It is an A.P. Thus, $50=0+(k-1)10$ $50=10k-10\Rightarrow k=6$ Thus, the six terms of the given expansion in which $x$ and $y$ are free from radical signs.