Question

The number of terms which are free from radical

signs in the expansion of $(y^{\frac{1}{5}}+x^{\frac{1}{10}}{)}^{55}$ is

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

Answer 1

The correct option is B

6

In the expansion of $(y^{\frac{1}{5}}+x^{\frac{1}{10}}{)}^{55}$, the general

term is

$T_{r+1}$ = ${}^{55}{C}_r\left(y^{\frac{1}{5}}{)}^{55-r}\right(x^{\frac{1}{10}}{)}^r$ = ${}^{55}{C}_r{y}^{11-\frac{r}{5}}{x}^{\frac{r}{10}}$.

This $T_{r+1}$ will be independent of radicals if the exponents $\frac{r}{5}$ and $\frac{r}{10}$ are integers, for

0 ≤ r ≤ 55 which is possible only when

r = 0, 10, 20, 30, 40, 50.

∴ There are six terms viz. $T_1,T_{11},T_{21},T_{31},T_{41},T_{51}$

which are independent of radicals.