A) \[\frac{^{20}{{C}_{10}}}{2}\]
B) \[\frac{1}{2}\,{{\,}^{20}}{{C}_{9}}+{{2}^{19}}\]
C) \[\frac{1}{2}\,{{\,}^{20}}{{C}_{10}}+{{2}^{21}}\]
D) \[\frac{1}{2}\,{{\,}^{20}}{{C}_{10}}+{{2}^{19}}\]
Correct Answer: D
Solution :
[d] Number of ways, \[S{{=}^{20}}{{C}_{10}}{{+}^{20}}{{C}_{9}}{{+}^{20}}{{C}_{8}}+...{{+}^{20}}{{C}_{0}}\] \[{{=}^{20}}{{C}_{10}}{{+}^{20}}{{C}_{11}}{{+}^{20}}{{C}_{12}}+...{{+}^{20}}{{C}_{20}}\] \[\Rightarrow \,\,\,\,\,\,2S{{=}^{20}}{{C}_{10}}+{{2}^{20}}\] \[\therefore \,\,\,\,S=\frac{^{20}{{C}_{10}}}{2}+{{2}^{19}}\]
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