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
B
1001C50
C
1002C50
D
1002C49

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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