A) \[^{31}{{C}_{0}}^{2}-{{\,}^{31}}{{C}_{1}}^{2}+{{\,}^{31}}{{C}_{2}}^{2}-...-{{\,}^{31}}{{C}_{31}}^{2}\]
B) \[^{32}{{C}_{0}}^{2}-{{\,}^{32}}{{C}_{1}}^{2}+{{\,}^{32}}{{C}_{1}}^{2}-...+{{\,}^{32}}{{C}_{32}}^{2}\]
C) \[^{32}{{C}_{0}}^{2}+{{\,}^{32}}{{C}_{1}}^{2}+{{\,}^{32}}{{C}_{2}}^{2}-..+{{\,}^{32}}{{C}_{32}}^{2}\]
D) \[^{34}{{C}_{0}}^{2}-{{\,}^{34}}{{C}_{1}}^{2}+{{\,}^{34}}{{C}_{2}}^{2}-...+{{\,}^{34}}{{C}_{32}}^{2}\]
Correct Answer: C
Solution :
[c] We know that \[^{n}{{C}_{0}}^{2}+{{\,}^{n}}{{C}_{1}}^{2}+...+{{\,}^{n}}{{C}_{n}}^{2}=2{{\,}^{n}}{{C}_{n}}\] and \[^{n}{{C}_{0}}^{2}-{{\,}^{n}}{{C}_{1}}^{2}+...+{{\,}^{n}}{{C}_{n}}^{2}\] \[\begin{align} & =\left\{ \begin{matrix} 0, \\ {{C}_{n/2}}{{(-1)}^{n/2}}, \\ \end{matrix} \right.\,\,\begin{matrix} if\,\,n\,\,is\,\,odd \\ f\,\,n\,\,is\,\,even \\ \end{matrix} \\ & \,\,\, \\ & ^{\,}i \\ \end{align}\] From this \[^{31}{{C}_{0}}^{2}-{{\,}^{31}}{{C}_{0}}^{2}+{{\,}^{31}}{{C}_{2}}^{2}-...-{{\,}^{31}}{{C}_{31}}^{2}=0\] \[^{32}{{C}_{0}}^{2}-{{\,}^{32}}{{C}_{1}}^{2}+{{\,}^{32}}{{C}_{2}}^{2}-...+{{\,}^{32}}{{C}_{32}}^{2}=-{{\,}^{32}}{{C}_{16}}\] \[^{34}{{C}_{0}}^{2}-{{\,}^{34}}{{C}_{1}}^{2}+{{\,}^{34}}{{C}_{2}}^{2}-...+{{\,}^{34}}{{C}_{32}}^{2}=-{{\,}^{34}}{{C}_{17}}\] \[^{32}{{C}_{0}}^{2}+{{\,}^{32}}{{C}_{1}}^{2}+{{\,}^{32}}{{C}_{2}}^{2}-...+{{\,}^{32}}{{C}_{32}}^{2}={{\,}^{64}}{{C}_{32}}\] Obviously \[^{64}{{C}_{32}}\] is greatest.
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