The number of rational terms in the expansion of (1+2+53)6 is

# The number of rational terms in the expansion of $\left(1+\sqrt{2}+\sqrt[3]{5}{\right)}^{6}$ is

1. A

7

2. B

11

3. C

12

4. D

13

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### Solution:

A term in the expansion  of $\left(1+\sqrt{2}+\sqrt[3]{5}{\right)}^{6}$ is of the form

$\frac{6!}{r!s!\left(6-r-s\right)!}\left(1{\right)}^{6-r-s}\left(\sqrt{2}{\right)}^{r}\left(\sqrt[3]{5}{\right)}^{s}$

$=\frac{6!}{r!s!\left(6-r-s\right)!}\left({2}^{r/2}\right)\left({5}^{s/3}\right)$

This term will be rational if ${2}^{r/2}$,and ${5}^{s/3}$ are both rational
numbers. This is possible if and only if r is a multiple of 2
and s is a multiple of 3. Possible values of r are 0, 2, 4 and
6 whereas the possible values of s are 0, 3 and 6. Also note that

For $0\le r+s\le 6$. s can take value 0, 3 or 6.

For $r=0$ s can take value 0 or 3.

For r = 4, s can take value 0.

For r = 6, s can take value 0.

Thus, the number of rational terms is 3 + 2 + 1 + 1 = 7

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