Question

The sum of 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

Answer: (B)

$\frac{1}{2}{{}^{20}C}_{10}$

$(1+x{)}^{20}{=}^{20}{C}_0{+}^{20}{C}_{1}x+....{+}^{20}{C}_{10}{x}^{10}+....{+}^{20}{C}_{20}{x}^{20}$

Put $x=-1$

$\Rightarrow 0{=}^{20}{C}_0{-}^{20}{C}_1+...{-}^{20}{C}_9{+}^{20}{C}_{10}{-}^{20}{C}_{11}+.....{+}^{20}{C}_{20}$

$\Rightarrow 0=2{(}^{20}{C}_0{-}^{20}{C}_1+....{-}^{20}{C}_9){+}^{20}{C}_{10}$

${\Rightarrow }^{20}{C}_{10}=2{(}^{20}{C}_0{-}^{20}{C}_1+....{-}^{20}{C}_9{+}^{20}{C}_{10})$

${\Rightarrow }^{20}{C}_0{-}^{20}{C}_1+....{+}^{20}{C}_{10}=\frac{1}{2}{}^{20}{C}_{10}$