A) \[0\]
B) \[1242\]
C) \[7315\]
D) \[6345\]
Correct Answer: A
Solution :
Key Idea: If \[n\] and \[r\]positive integer and\[0\le r<n\], then \[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{n+1}}{{C}_{r}}\] \[\therefore \] \[^{20}{{C}_{4}}+{{2.}^{20}}{{C}_{3}}{{+}^{20}}{{C}_{2}}{{-}^{22}}{{C}_{18}}\] \[={{(}^{20}}{{C}_{4}}{{+}^{30}}{{C}_{3}})+{{(}^{20}}{{C}_{3}}{{+}^{20}}{{C}_{2}}){{-}^{22}}{{C}_{18}}\] \[{{=}^{21}}{{C}_{4}}{{+}^{21}}{{C}_{3}}{{-}^{22}}{{C}_{18}}\] \[{{=}^{22}}{{C}_{4}}{{-}^{22}}{{C}_{18}}\] \[{{=}^{22}}{{C}_{18}}{{-}^{22}}{{C}_{18}}\] \[=0\]
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