Find the coefficient of x18 in x 2 - 3a x 15-

We can write the expression as x^30 ( 1+3ax^3 )^15 which can be written as x^30∑_r=0 ^15 ^15C_r #160; ( 3ax^3)^15 r Coefficient of x^18 means ...

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Standard XII

Mathematics

Binomial Coefficients

Find the coef...

Question

Find the coefficient of $x^{18}$ in ${(x^{2} + \frac{3 a}{x})}^{15}$ .

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Solution

We can write the expression as $x^{30} {(1 + \frac{3 a}{x^{3}})}^{15}$ which can be written as $x^{30} 15 \sum r = 0^{15} C_{r} {(\frac{3 a}{x^{3}})}^{15 - r}$

Coefficient of $x^{18}$ means $30 - 3 (15 - r) = 18 \Rightarrow r = 11$

The coefficient of $x^{18}$ is $^{15} C_{11} (3 a)^{4}$

$\Rightarrow 110565 a^{4}$

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Find the coefficient of x18 in (x2+3ax)15.

We can write the expression as x30(1+3ax3)15 which can be written as x3015∑r=0 15Cr(3ax3)15−rCoefficient of x18 means 30−3(15−r)=18⇒r=11The coefficient of x18 is 15C11(3a)4⇒110565a4

Find the coefficient of $x^{18}$ in ${(x^{2} + \frac{3 a}{x})}^{15}$ .