Question

The number of term with integeral coefficients in the expansion of $({17}^{\frac{1}{3}}+{35}^{\frac{1}{2}}x{)}^{600}$ is

• A. 100
• B. 50
• C. 150
• D. 101

Answer 1

The correct option is D

101

101

The general term $T_{r+1}$ in the given expansion is given by

${}^{600}{C}_r{\left({17}^{\frac{1}{3}}\right)}^{600-r}{\left({35}^{\frac{1}{2}}x\right)}^r$

Now, $T_{r+1}$ is an integer if $\frac{r}{2}$ and $\frac{r}{3}$ are integers for all $0\le r\le 600$

Thus, we have

r =0, 6,12,....600 (Multiplies of 6)

Since, It is an A.P

So, 600=0+(n-1)6

$\Rightarrow n=101$

Hence, there are 101 terms with integral coefficients.