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Formation of Algebraic Expressions

Coefficient o...

Question

Coefficient of $x^{n}$ in the expression ${(1 - 2 x + 3 x^{2} - 4 x^{3} + . . .)}^{- n}$ equals

{^{2 n} C}_{n}

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{^{2 n + 1} C}_{n}

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{^{3 n} C}_{n}

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0

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Solution

The correct option is A ${^{2 n} C}_{n}$
$(1 - 2 x + 3 x^{2} - 4 x^{2} . . . \infty)^{- n}$
$= [(1 + x)^{- 2}]^{- n}$
$= (1 + x)^{2 n}$
Hence the coefficient of $x^{n}$ will be
$^{2 n} C_{n}$

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