Find the number of terms which are free from radical signs in the expansion of `(y^(1//5)+x^(1//10))^(55)dot`

Explanation

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}} 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`.


`therefore` There are six terms viz. `{T}_{1}, {~T}_{11}, {~T}_{21}, {~T}_{31}, {~T}_{41}, {~T}_{51}`, which are independent of radicals.