Question

Coefficient of x^11 in (2x^2+3/x^3)^13

Answer 1

Answer:

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Answer 2

SOLUTION

FORMULA TO BE IMPLEMENTED

The ${ \: (r + 1)}^{th}$term of the Binomial Expansion of ${(x + a)}^{n}$ is $\displaystyle \: T_{r+1} = \binom{n}{r} \: {a}^{n - r} {x}^{r}$

Coefficient of $\displaystyle \: {x}^{r} \: \: is \: \: \binom{n}{r} \: {a}^{n - r}$

EVALUATION

Let ${x}^{11}$appears in ${ \: (r + 1)}^{th}$term of the Binomial Expansion of $\displaystyle \: {(2 {x}^{2} + \frac{3}{ {x}^{3} }) }^{13}$

Then

$\displaystyle \: T_{r+1} = \binom{13}{r} \: {(2 {x}^{2}) }^{13 - r} {( \frac{3}{ {x}^{3} } )}^{r}$

$\implies \: \displaystyle \: T_{r+1} = \binom{13}{r} \: {2}^{13 - r} \times {3}^{r} \times {( {x}) }^{26 - 2r - 3r}$

$\implies \: \displaystyle \: T_{r+1} = \binom{13}{r} \: {2}^{13 - r} \times {3}^{r} \times {( {x}) }^{26 - 5r}$

Therefore

$26 - 5r \: = 11$

$\implies \: - 5r = - 15$

$\therefore \: \: r \: = 3$

Hence the required coefficient is

$\displaystyle \: = \binom{13}{3} \: \times {2}^{13 - 3} \times {3}^{3}$

$\displaystyle \: = \binom{13}{3} \: \times {2}^{10} \times {3}^{3}$

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