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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

In the expansion of \left(y^{1 / 5}+x^{1 / 10}\right)^{55}, the general term is T_{r+1}={ }^{55} C_r\left(y^{1 / 5}\right)^{55-r}\left(x^{1 / 10}\right)^r={ }^{55} C_r \cdot 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 \leq r \leq 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.

Posted by

Devendra Khairwa

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