Question

Find the number of integral terms in the expansion of ${\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)}^{120}$.

• A. $1$
• B. $21$
• C. $20$
• D. $11$

Answer 1

The correct option is B

$21$

Explanation for the correct answer:

Calculating the number of integral terms in this given expansion.

We know, $(r+1)$ th term of the expansion ${\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)}^{120}$is

$$\begin{gathered} =T_{r+1} \\ =C_r^{120}{\left(4\right)}^{\frac{120-r}{4}}{\left(5\right)}^{\frac{r}{6}} \end{gathered}$$

Therefore, $r$ must be even integral of $6$and ranges from $0$ to $120.$

Hence, $r=0,6,12,.....,120$

Total integral terms in that given expansion is $21$.

Therefore, Option(B) is correct.