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Question

Coefficient of $$x^{50}$$, 
$$(x>0)$$, in $$(1+{x})^{1000}+2{x}(1+{x})^{999}+{3}{{x}}^{2}(1+{x})^{998}+\ldots$$ is


A
1000C50
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B
1001C50
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C
1002C50
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D
1002C49
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Solution

The correct option is C $$^{1002}C_{50}$$

$${ (1+x) }^{ 1000 }+2x{ (1+x) }^{ 999 }+3{ x }^{ 2 }{ (1+x) }^{ 998 }... { \Rightarrow (1+x) }^{ 1000 }(1+2\left( \dfrac  { x }{ x+1 }  \right) +3{ \left( \dfrac  { x }{ x+1 }  \right)  }^{ 2 }+...)$$ 
Considering $$ \left| \dfrac  { x }{ x+1 }  \right| <1$$, we get
Given expression $$= (1+x)^{ 1000 }{ \left( 1-\dfrac  { x }{ x+1 }  \right)  }^{ -2 }={ (1+x) }^{1002}$$ Coefficient of $${ x }^{ 50 }$$ is $$ _{  }^{ 1002 }{ { C }_{ 50 } }$$.


Mathematics

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