Question

The sum of the series ${}^{20}{C}_0-$ ${}^{20}{C}_1+$ ${}^{20}{C}_2-$ ${}^{20}{C}_3+$ ______$-$ __________$+{}^{20}{C}_{10}$ is?

• A. $-$ ${}^{20}{C}_{10}$
• B. $\frac{1}{2}$ ${}^{20}{C}_{10}$
• C. $0$
• D. ${}^{20}{C}_{10}$

Answer 1

The correct option is C $0$

${}^{20}{C}_0{-}^{20}{C}_1{+}^{20}{C}_2{-}^{20}{C}_3+.........{.}^{20}{C}_{10}{\Rightarrow }^{20}{C}_0{+}^{20}{C}_2{-}^{20}{C}_4+.........{.}^{20}{C}_{10}-\left({}^{20}{C}_1{+}^{20}{C}_3{-}^{20}{C}_5+......{+}^{20}{C}_9\right)\because n=20\Rightarrow 2^{\left(20-1\right)}-\left(2^{\left(20-1\right)}\right)\left[{\because }^n{C}_0{-}^n{C}_2{+}^n{C}_4{=}^n{C}_1{-}^n{C}_3{+}^n{C}_5=2^{\left(n-1\right)}\right]=0$