Question

Number of irrational terms in the expansion of ${(\sqrt{2}+\sqrt{3})}^{15}$ is equal to

• A. $16$
• B. $7$
• C. $12$
• D. $15$

Answer 1

The correct option is A $16$

In the expansion of $(\sqrt{2}+\sqrt{3}{)}^{15}$

Total Number of termed $=15+1=16$

General term $=T_{n+1}={}^{15}{C}_r\left(\sqrt{2}{)}^{15-r}\right(\sqrt{3}{)}^r$

${}^{15}{C}_r{2}^{\frac{15-r}{2}}{3}^{\frac{r}{2}}$

For the terms to be rational $\frac{r}{2}$ and $\frac{15-r}{2}$ should be integers

For $\frac{r}{2}$ to be integer : $r=0,2,4,6,8,10,12,14$

For $\frac{15-r}{2}$ to be integer : $r=1,3,5,7,9,11,13,15$

$\Rightarrow$ No common value of $r$ to make both exponents to be integers

$\Rightarrow$ No rational term will be there i.e. all the terms will be irrational

$\Rightarrow$ Number of irrational terms $=16$