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Ta có:
$(1+x)^n =\sum_{i=0}^nC^i_nx^i$
$\Rightarrow n(1+x)^{n-1}=\sum_{i=1}^{n}iC^i_nx^{i-1}$
$\Rightarrow nx(1+x)^{n-1}=\sum_{i=1}^{n}iC^i_nx^i $
$\Rightarrow n(1+x)^{n-1}+n(n-1)x(1+x)^{n-2}=\sum_{i=1}^ni^2C^i_nx^{i-1}$
Thay $x=1$, ta được:
$ S=1^2C^1_n+2^2C^2_n+3^2C^3_n+....+n^2C^n_n=n.2^{n-1}+n(n-1)2^{n-2}$
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