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Greatest Binomial Coefficients

Find the coef...

Question

Find the coefficient of:
(i) $x$ in the expansion of $(1 - 2 x^{3} + 3 x^{5}) {(1 + \frac{1}{x})}^{8}$

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Solution

Coeff of x in $(1 - 2 x^{3} + 3 x^{5}) (1 + \frac{1}{x})^{8}$

= coeff of x in $(1 + \frac{1}{x})^{8} +$ coeff of x in $(- 2 x^{3}) (1 + \frac{1}{x})^{8} +$ coeff of x in $3 x^{5} (1 + \frac{1}{x})^{8}$

= coeff of x in $(1 + \frac{1}{x})^{8} - 2$ coeff of $x^{- 2}$ in $(1 + \frac{1}{x})^{8} + 3$ coeff of $x^{- 4}$ in $(1 + \frac{1}{x})^{8}$

$r^{t h}$ term in $(1 + \frac{1}{x})^{8}$

$t_r = {^8C}_{r-1} ( \frac{1}{x})^{r-1} = {^8C_{r-1} x ^{1-r} $

For $1 - r = 1, r = 0,$ coeff of $x = 0$

For $1 - r = - 2, r = 3,$ coeff of $x = {^{8} C}_{2}$

For $1 - r = - 4, r = 5,$ coeff of $x = {^{8} C}_{4}$

Therefore coeff of $x = 0 - 2 {^{8} C}_{2} + 3 \times {^{8} C}_{4}$
$= 154$

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$(i) x^{10}$ in the expansion of ${(2 x^{2} - \frac{1}{x})}^{20}$

$(i i) x^{7}$ in the expansion of ${(x - \frac{1}{x^{2}})}^{40}$

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$(i v) x^{9}$ in the expansion of ${(x^{2} - \frac{1}{3 x})}^{9}$

$(v) x^{m}$ in the expansion of ${(x + \frac{1}{x})}^{n}$

(vi)x in the expansion of $(1 - 2 x^{3} + 3 x^{5}) {(1 + \frac{1}{x})}^{8}$

$(v i i) a^{5} b^{7}$ in the expansion of $(a - 2 b)^{12} .$

(viii)x in the expansion of $(1 - 3 x + 7 x^{2}) (1 - x)^{16}$

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{(3 x^{2} - \frac{a}{3 x^{3}})}^{10}

x^{9}

{(x^{2} - \frac{1}{3 x})}^{9}

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(1 - 2 x^{3} + 3 x^{5}) {(1 + \frac{1}{x})}^{8}

a^{5} b^{7}

{(a - 2 b)}^{12}

(1 - 3 x + 7 x^{2}) {(1 - x)}^{16}

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