The coefficient of x - 6 in the expansion of ( 4 x 5 + 5 2 x 2 ) 9 is

Question

The coefficient of $x^{- 6}$ in the expansion of ${(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}$ is _________.

Smart Achievers · Accepted Answer

(5040)

$Here,$ $T_{r + 1} = {}^{9}C_{r} {(\frac{4 x}{5})}^{9 - r} {(\frac{5}{2 x^{2}})}^{r}$

$= {}^{9}C_{r} {(\frac{4}{5})}^{9 - r} {(\frac{5}{2})}^{r} x^{9 - r - 2 r}$

$For coefficient of x^{- 6}, we have 9 - r - 2 r = - 6 \Rightarrow 3 r = 15$

$\Rightarrow$ $r = 5$

$So, coefficient of x^{- 6} = {}^{9}C_{5} {(\frac{4}{5})}^{4} {(\frac{5}{2})}^{5}$

$= \frac{9!}{5! 4!} (\frac{4 \times 4 \times 4 \times 4}{5 \times 5 \times 5 \times 5}) (\frac{5 \times 5 \times 5 \times 5 \times 5}{2 \times 2 \times 2 \times 2 \times 2})$

$= \frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2} (8) (5) = 126 \times 40$

$∴$ $126 \times 40 \cdot x^{- 6} = 5040$

Solution

Q.
The coefficient of $x^{- 6}$ in the expansion of ${(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}$ is _________.

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Solution

Q. The coefficient of x-6 in the expansion of (4x5+52x2)9 is _________.

Ans. (5040) Here, Tr+1=Cr9(4x5)9-r(52x2)r =Cr9(45)9-r(52)rx9-r-2r For coefficient of x-6, we have 9-r-2r=-6⇒3r=15 ⇒ r=5 So, coefficient of x-6=C59(45)4(52)5 =9!5! 4!(4×4×4×45×5×5×5)(5×5×5×5×52×2×2×2×2) =9·8·7·64·3·2(8)(5)=126×40 ∴ 126×40·x-6=5040

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Q.
The coefficient of $x^{- 6}$ in the expansion of ${(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}$ is _________.