Coefficient of x 1007 in 1- x 2006- x 1- x 2005- x 21- x 2004- x 2006 isA- 2006 C 1008B- 2007 C 1007- 2007 C 1008D- 2006 C 1007

The correct option is B ^2007C_1007 S=(1+x)^2006 ( 1 ( x/1+x )^2007 )/1 x/1+x =(1+x)^2006 ( 1 ( x/1+x )^2007 )/1/1+x =(1+x)^2007 ( 1 ( x/1+x )^2007 ) ⇒ (1 ...

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Standard XII

Mathematics

General Term of Binomial Expansion

Coefficient o...

Question

Coefficient of $x^{1007}$ in
$(1 + x)^{2006} + x (1 + x)^{2005} + x^{2} (1 + x)^{2004} + \dots + x^{2006}$ is

$^{2006} C_{1007}$

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$^{2006} C_{1008}$

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$^{2007} C_{1007}$

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$^{2007} C_{1008}$

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Solution

The correct option is C
$^{2007} C_{1007}$

$S = \frac{(1 + x)^{2006} (1 - {(\frac{x}{1 + x})}^{2007})}{1 - \frac{x}{1 + x}} = \frac{(1 + x)^{2006} (1 - {(\frac{x}{1 + x})}^{2007})}{\frac{1}{1 + x}} = (1 + x)^{2007} (1 - {(\frac{x}{1 + x})}^{2007}) \Rightarrow (1 + x)^{2007} - \frac{x^{2007}}{1}$
$\Rightarrow$ Coefficient of $x^{1007} =^{2007} C_{1007}$

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Coefficient of x1007 in (1+x)2006+x(1+x)2005+x2(1+x)2004+⋯+x2006 is

The correct option is C 2007C1007 S=(1+x)2006(1−(x1+x)2007)1−x1+x=(1+x)2006(1−(x1+x)2007)11+x=(1+x)2007(1−(x1+x)2007)⇒(1+x)2007−x20071 ⇒ Coefficient of x1007=2007C1007

Coefficient of $x^{1007}$ in
$(1 + x)^{2006} + x (1 + x)^{2005} + x^{2} (1 + x)^{2004} + \dots + x^{2006}$ is