Find the coefficient of $x^{4}$ in the expression of $(1 - x)^{12} . (1 + 2 x)^{6}$ .

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Solution

$(1 - x)^{12} ↓ (1 + 2 x)^{6} ↓$
$x^{4} \to x x^{3}$ ...(1)
$x^{2} x^{2}$ ...(2)
$x^{3} x$ ...(3)
$x^{4} x^{0}$ ...(4)
$x^{0} x^{4}$ ...(5)
(1) $[^{12} C_{1} (1)^{11} (- x)^{1}] \times [^{6} C_{3} (1)^{6 - 3} (2 x)^{3}] = (- 12 x) (20 \times 8 x^{3}) = - 12 x \times 160 x^{3} = - 1920 x^{4}$
(2) $[^{1} 2 C_{2} (1)^{12 - 2} (- x)^{2}] [^{6} C_{2} (1)^{6 - 2} (2 x)^{2}] = (66 x^{2}) (15 \times 4 x^{2}) = 3960 x^{4}$
(3) $[^{12} C_{3} (1)^{12 - 3} (- x)^{3}] [^{6} C_{1} (1)^{6 - 1} (2 x)^{1}] = (- 220 x^{3}) (12 x) = - 2640 x^{4}$
(4) $[^{12} C_{4} (1)^{12 - 4} (- x)^{4}] [^{6} C_{0} (1)^{6 - 0} (2 x)^{0}] = (495 x^{4}) (1) = 495 x^{2}$
(5) $[^{12} C_{0} (1)^{12 - 0} (- x)^{0}] [^{6} C_{4} (1)^{6 - 4} (2 x)^{4}] = 1 \times (15 \times 16 x^{4}) = 240 x^{4}$
Coefficient of $x^{4}$ in the expression = Sum of coefficient of $x^{4}$ in (1), (2), (3), (4) and (5)
= -1920 + 3960 - 2640 + 495 + 240
= 2040 - 2145 + 240
= -105 + 240
= 135

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Similar questions

Find the coefficient of $x^{5}$ in ${(1 + 2 x)}^{6} + {(1 - x)}^{7}$ .

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x^{160}

(x^{8} + 1)^{60} {(x^{12} + 3 x^{4} + \frac{3}{x^{4}} + \frac{1}{x^{12}})}^{- 10}

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{(1 + 2 x)}^{6} {(1 - x)}^{7}

f (x) = 2 x^{6} + x^{4} - 1

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