First slide

Binomial theorem for positive integral Index

Question

If the $(r + 1) t h$ term in the expansion of ${(\frac{a^{1 / 3}}{b^{1 / 6}} + \frac{b^{1 / 2}}{a^{1 / 6}})}^{21}$ has equal exponents of both $a$ and $b$ , then value of $r$ is

Moderate

A

8
B

9
C

10
D

11

Solution

We have
$\begin{array}{l} T_{r + 1} =^{21} C_{r} {(\frac{a^{1 / 3}}{b^{1 / 6}})}^{21 - r} {(\frac{b^{1 / 2}}{a^{1 / 6}})}^{r} \\ =^{21} C_{r} \frac{a^{7 - r / 3}}{b^{7 / 2 - r / 6}} \cdot \frac{b^{r / 2}}{a^{r / 6}} =^{21} C_{r} a^{7 - r / 2} b^{2 / 3 - 7 / 2} \end{array}$
Since exponents of $a$ and $b$ in the $(r + 1) t h$ term are equal.
$7 - \frac{r}{2} = \frac{2 r}{3} - \frac{7}{2} \Rightarrow \frac{21}{2} = \frac{7}{6} r \Rightarrow r = 9 .$

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