Question

The number of terms which are free from radical signs in the expansion of $(y^{\frac{1}{5}}+x^{\frac{1}{10}}{)}^{55}$ are

• A. 5
• B. 6
• C. 7
• D. none of these

Answer 1

The correct option is B 6

In the expansion of $(y^{1/5}+x^{1/10}{)}^{55}$,the general term is
$T_{r+1}{=}^{55}{C}_r\left(y^{1/5}{)}^{55-r}\right(x^{1/10}{)}^r{=}^{55}{C}_r.y^{11-r/5}{x}^{r/10}$
This $T_{r+1}$ will be independent of radicals if the exponents r/5 and r/10 are integers, for $0\le r\le 55$ which is possible only when $r=0,10,20,30,40,50$.
$\therefore$ There are six terms viz. $T_1,T_{11},T_{21},T_{31},T_{41},T_{51}$, which are independent of radicals.