First slide

Binomial theorem for positive integral Index

Question

If coefficients of $r$ th and $(r + 1)$ th term in the expansion of ${(3 + 2 x)}^{74}$ are equal, then $r$ is equal to:

Moderate

A

28
B

29
C

30
D

31

Solution

$T_{r + 1} =^{74} C_{r} 3^{74 - r} (2 x)^{r}$
$\Rightarrow$ coeffcient of $(r + 1)$ th term is $^{74} C_{r} 3^{74 - r} 2^{r}$
We are given $^{74} C_{r - 1} 3^{74 - r + 1} 2^{r - 1} =^{74} C_{r} 3^{74 - r} 2^{r}$
$\begin{array}{l} \Rightarrow \frac{3}{2} = \frac{74!}{r! (74 - r)!} \cdot \frac{(r - 1)! (75 - r)!}{74!} \\ = \frac{75 - r}{r} \\ \Rightarrow 3 r = 150 - 2 r \Rightarrow r = 30 \end{array}$

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