A) 5
B) 6
C) 7
D) None of these
Correct Answer: B
Solution :
In the expansion of \[{{({{y}^{1/5}}+{{x}^{1/10}})}^{55}}\], the general term is \[{{T}_{r+1}}={{\,}^{55}}{{C}_{r}}{{({{y}^{1/5}})}^{55-r}}{{({{x}^{1/10}})}^{r}}={{\,}^{55}}{{C}_{r}}.{{y}^{11-r/5}}{{x}^{r/10}}\]. 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\[r=0,10,\,20,\,30,\,40,50\]. \ There are six terms viz. \[{{T}_{1}},{{T}_{11}},{{T}_{21}},{{T}_{31}},{{T}_{41}},{{T}_{51}}\] which are independent of radicals.
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