what-is-coefficient-of-x-2-in-the-expansion-1-x-1-2x-6-

Question Number 85817 by jagoll last updated on 25/Mar/20

what is coefficient of x^{2} in

the expansion {[(1 - x) (1 + 2 x)]}^{6}

Answered by john santu last updated on 25/Mar/20

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

\underset{a + b + c ⩽ 6}{\sum^{6}} (\begin{matrix} 6 \\ a, b, c \end{matrix}) 1^{a} x^{b} {(- 2 x^{2})}^{c}

\Rightarrow a + b + 2 c = 6

(i) c = 1 \land a = 5

\Rightarrow \frac{6!}{5! . 1!} . 1^{5} {(- 2 x^{2})}^{1} = - 12 x^{2}

(i i) b = 2 \land a = 4

\Rightarrow \frac{6!}{4! . 2!} 1^{4} . x^{2} = 15 x^{2}

h e n c e c o e f f i c i e n t x^{2} = - 12 + 15 = 3

Commented by jagoll last updated on 25/Mar/20

thank you sir

Answered by mr W last updated on 25/Mar/20

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

c o e f . o f x^{2} = \frac{6 \times 5}{2} + 6 \times (- 2) = 3

Commented by jagoll last updated on 25/Mar/20

this is short cut sir ?

Commented by mr W last updated on 25/Mar/20

e a c h o f t h e 6 x^{'} s w i l l b e m u l t i p l i c a t e d

e a c h o f t h e o t h e r x^{'} s t o f o r m x^{2},

t h e r e a r e C_{2}^{6} c o m b i n a t i o n s, s u m o f

t h e c o e f . i s \frac{6 \times 5}{2} .

e a c h o f t h e 6 (- 2 x^{2}) w i l l b e m u l t i p l i c a t e d

t h e o t h e r 1^{'} s, t h e s u m o f t h e c o e f . i s

6 \times (- 2) .

t h e r e f o r e \Rightarrow \frac{6 \times 5}{2} + 6 \times (- 2) = 3 .

Commented by jagoll last updated on 25/Mar/20

in generally sir

if the question {(1 + x - {2 x}^{2})}^{n}

for n even ?

Commented by jagoll last updated on 25/Mar/20

C_{2}^{n} + (- 2 n) sir

Commented by mr W last updated on 25/Mar/20

n i s a n y + v e i n t e g e r, {m u s t n}^{'} t b e e v e n .

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