Maths Question:
$\text{Prove that}{{\text{ }}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}+--{{-}^{n}}{{C}_{n}}={{2}^{n}}$
Maths Solution:
$\begin{align} & ^{n}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}+--{{-}^{n}}{{C}_{n}}={{2}^{n}} \\ & {{(1+x)}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}x{{+}^{n}}{{C}_{2}}{{x}^{2}}{{+}^{n}}{{C}_{3}}{{x}^{3}}+--{{-}^{n}}{{C}_{n}} \\ & \text{Let }x=1 \\ & {{(1+1)}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}+{{+}^{n}}{{C}_{3}}+--{{-}^{n}}{{C}_{n}} \\ & {{2}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}+{{+}^{n}}{{C}_{3}}+--{{-}^{n}}{{C}_{n}} \\\end{align}$
University mathstopic: