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$(1+x)^n-nx-1$ is divisible by ?

$\begin{array}{1 1} x^2 \\ x^3 \\ x^4 \\ x^5 \end{array} $

1 Answer

$(1+x)^n =^nC_0+^nC_1x+^nC_2x^2+......^nC_nx^n$
$\therefore\:$$(1+x)^n-nx-1=\big(1+nx+^nC_2x^2+^nC_3x^3+.......\big)-nx-1$
$=^nC_2x^2+^nC_3x^3+......^nC_nx^n$
which is divisible by $x^2$.
answered Oct 16, 2013 by rvidyagovindarajan_1
 

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