A) \[{{4}^{50}}\]
B) \[{{2}^{50}}\]
C) \[{{2}^{101}}-1\]
D) \[{{2}^{101}}\]
Correct Answer: A
Solution :
\[(x+{{\,}^{101}}{{C}_{0}})(x+{{\,}^{101}}{{C}_{0}})...(x+{{\,}^{101}}{{C}_{50}})\]contains 51 linear factors. Thus, x50 is obtained by multiplying \[x's\] from any 50 factors and the constant term from the remaining one factor. Hence, the coefficient of\[{{x}^{50}}=S={{\,}^{101}}{{C}_{0}}{{+}^{101}}{{C}_{1}}+...\]\[+\,{{\,}^{101}}{{C}_{50}}\] Noting that \[{{\,}^{101}}{{C}_{51}}={{\,}^{101}}{{C}_{50}},{{\,}^{101}}{{C}_{101}}={{\,}^{101}}{{C}_{0}}\]etc., We see that \[2S={{\,}^{101}}{{C}_{0}}{{+}^{101}}{{C}_{1}}+...{{+}^{101}}{{C}_{50}}{{+}^{101}}{{C}_{51}}+...{{+}^{101}}{{C}_{101}}\] \[={{2}^{101}}\] \[\Rightarrow \] \[S={{2}^{100}}={{4}^{50}}\]
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