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  • The number of terms which are free from radical signs in the expansion of <img alt="\left(y^{1 / 5}+x^{1 / 10}\right)^{55}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%28y%5E%7B1%2

The number of terms which are free from radical signs in the expansion of \left(y^{1 / 5}+x^{1 / 10}\right)^{55} is:

Option: 1

5


Option: 2

6


Option: 3

7


Option: 4

None of these


Answers (1)

best_answer

The general term in the expansion of \left(y^{1 / 5}+x^{1 / 10}\right)^{55} is
$$ T_{r+1}={ }^{55} C_r\left(y^{1 / 5}\right)^{55-r}\left(x^{1 / 10}\right)^r={ }^{55} C_r y^{11-r / 5} x^{r / 10}
Clearly, T_{r+1}will be independent of radical signs if r / 5 and r / 10  are integers, where 0 \leq r \leq 55
\therefore \quad r=0,10,20,30,40,50
Hence, there are 6 terms in the expansion of the given expression which are independent of radicals.

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