A) \[n{{(n-1)}^{n-2+n{{.2}^{n-1}}}}\]
B) \[n{{(n-1)}^{n-2}}\]
C) \[n{{(n-1)}^{n-3}}\]
D) None of these
Correct Answer: D
Solution :
[d] We know, \[\sum\limits_{r=1}^{n}{{{r}^{2}}{{.}^{n}}{{C}_{r}}=n(n-1)}{{2}^{n-2}}+n{{.2}^{n-1}}\] ? (1) and \[\sum\limits_{r=1}^{n}{{{(-1)}^{r-1}}.{{r}^{2}}{{.}^{n}}{{C}_{r}}=0}\] ? (2) Adding (1) & (2) we get, \[2[{{1}^{2}}.{{C}_{1}}+{{3}^{2}}.{{C}_{3}}+{{5}^{2}}{{C}_{5}}+...]=n(n-1){{2}^{n-2}}+n{{.2}^{n-1}}\] \[\Rightarrow [{{1}^{2}}{{C}_{1}}+{{3}^{2}}{{C}_{3}}+{{5}^{2}}{{C}_{5}}+...]\]\[=n(n-1){{2}^{n-3}}+n{{.2}^{n-2}}.\]
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