A) \[^{31}{{C}_{20}}{{-}^{21}}{{C}_{10}}\]
B) \[^{31}{{C}_{21}}\]
C) \[^{31}{{C}_{20}}\]
D) None of these
Correct Answer: A
Solution :
[a] The required number of ways \[{{=}^{21}}{{C}_{10}}{{+}^{22}}{{C}_{10}}{{+}^{23}}{{C}_{10}}+...{{+}^{30}}{{C}_{10}}\] \[={{(}^{21}}{{C}_{10}}{{+}^{21}}{{C}_{11}}{{+}^{22}}{{C}_{12}}{{+}^{23}}{{C}_{13}}+...{{+}^{30}}{{C}_{20}}){{-}^{21}}{{C}_{11}}\] \[{{=}^{22}}{{C}_{11}}{{+}^{22}}{{C}_{12}}{{+}^{23}}{{C}_{13}}+...{{+}^{30}}{{C}_{20}}{{-}^{21}}{{C}_{10}}\] \[({{\because }^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r}}{{=}^{n+1}}{{C}_{r}})\] \[{{=}^{31}}{{C}_{20}}{{-}^{21}}{{C}_{10}}.\]
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