The number of terms which are free from radic | vedclass

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(B) In the expansion of $(y^{1/5} + x^{1/10})^{55}$,the general term is given by:$T_{r+1} = {}^{55}C_r (y^{1/5})^{55-r} (x^{1/10})^r = {}^{55}C_r \cdo

Vedclass · Accepted Answer

$6$

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(B) In the expansion of $(y^{1/5} + x^{1/10})^{55}$,the general term is given by:$T_{r+1} = {}^{55}C_r (y^{1/5})^{55-r} (x^{1/10})^r = {}^{55}C_r \cdot y^{11 - r/5} \cdot x^{r/10}$.For the term to be free from radical signs,the exponents of $x$ and $y$ must be integers.This requires $r/5$ and $r/10$ to be integers.Since $0 \le r \le 55$,$r$ must be a multiple of $10$ (the least common multiple of $5$ and $10$).The possible values for $r$ are $0, 10, 20, 30, 40, 50$.There are $6$ such values,corresponding to the terms $T_1, T_{11}, T_{21}, T_{31}, T_{41}, T_{51}$.Thus,there are $6$ terms free from radical signs.

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

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