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The coefficient of $x^5$ in the expansion of $(1+x)^{21}$ + $(1+x)^{22}$ + ...... + $(1+x)^{30}$ is

$\begin{array}{1 1} =^{30}C_5 +^{20} C_5 \\ =^{51} C_5 \\ =^{31} C_6 -^{21} C_6 \\ =^{30}C_5 -^{20} C_5 \end{array} $

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$\therefore\:$ Coeff. of $x^5$ in this expansion=
Coeff. of $x^6$ in the expansion $\big[(1+x)^{31}-(1+x)^{21}\big]$
answered Oct 14, 2013 by rvidyagovindarajan_1

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