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Find the coefficient of x^{11} in (2x^2+{3}{x^3})^{13}

Question

Find the coefficient of $x^{11}$ in $(2 x^{2} + \frac{3}{x ^{3}})^{13}$

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Find the coefficient of

x^{11}

in

(2 x^{2} + \frac{3}{x ^{3}})^{13}

11 mins ago

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11 mins ago

Text solutionVerified

(2 x^{2} + \frac{3}{x ^{3}})^{13}

\Rightarrow (r + 1)^{t h}

term

=^{13} C_{r} (2 x^{2})^{13 - r} (\frac{3}{x ^{3}})^{r}

\Rightarrow^{13} C_{r} 2^{13 - r} \times 3^{r} \times x^{26 - 5 r}

As we want

x^{11}

so,

x^{26 - 5 r} = x^{11}

\Rightarrow x^{26 - 5 r} = x^{11}

i.e,

26 - 5 r = 11

\Rightarrow 5 r = 15

\Rightarrow r = 3

So,

^{13} C_{3} \times 2^{13 - 3} \times 3^{3}

\Rightarrow \frac{13 !}{10 ! 3 !} \times 2^{10} \times 27

\Rightarrow \frac{11 \times 12 \times 13}{6} \times 2^{10} \times 27

\Rightarrow 7722 \times 2^{10} .

Hence, the answer is

7722 \times 2^{10} .

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Question Text	Find the coefficient of $x^{11}$ in $(2 x^{2} + \frac{3}{x ^{3}})^{13}$
Answer Type	Text solution:1
Upvotes	150