Question

The number of rational terms in the expansion of ${(x^{\frac{1}{2}}+y^{\frac{1}{3}})}^{18}$ is

• A. 5
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is D 4

Given, ${(x^{\frac{1}{2}}+y^{\frac{1}{3}})}^{18}$

General term, $T_{r+1}={}^n{C}_r{x}^{n-r}{y}^r$

$T_{r+1}={}^{18}{C}_r{\left(x^{\frac{1}{2}}\right)}^{18-r}{\left(y^{\frac{1}{3}}\right)}^r$

$T_{r+1}={}^{18}{C}_r\left(x{)}^{9-\frac{r}{2}}\right(y{)}^{\frac{r}{3}}$

To find the number of rational terms.

$r=0,6,12,18$

greater that $18$ we don't need to check because $n>r$

Hence, $4$ rational terms are possible at $r=0,6,12,18$

Hcen, option D is correct.