Question

Number of rational terms in expansion of $(\sqrt{2}+\sqrt{3}+\sqrt[3]{35}{)}^{20}$ is a two digit number ab, then a + b=

• A. 3
• B. 6
• C. 8
• D. 9

Answer 1

Answer: (C)

8

Given expansion is ${(\sqrt{2}+\sqrt{3}+\sqrt[3]{35})}^{20}$,

From multinomial theorem, the general term is ,

$\frac{20!}{p!q!r!}{2}^{\frac{p}{2}}{3}^{\frac{q}{2}}{5}^{\frac{r}{3}}$ where $p+q+r=20$ and $p,q,r$ are natural numbers,

For the term to be rational $p,q$ are multiples of $2$ and $r$ is a multiple of $3$,

if $r=0⟹$the number of solutions of the equation $p+q=20$ as $p,q$ are even, the number of solutions are equal to $x+y=10$ where $p=2x$ and $q=2y$

$\therefore$ the number of solutions are $11$,

Here $r$ cant be an odd multiple of $3$ because sum of two even numbers cant be odd,

$\therefore$ the next case is $r=6$,The number of values of $p,q$ are the solutions of the equation $x+y=7$ which are 8,

The next case is $r=12$,The number of values of $p,q$ are the solutions to the equation $x+y=4$ which are $5$,

The next case is $r=18$,the number of values of $p,q$ are the solutions to the equation $x+y=1$ which are $2$,

$\therefore$ the total number of rational terms are $26$.

The sum of the digits is $8$.