Question

${}^n{C}_0{+}^n{C}_2{+}^n{C}_4+\dots \cdots {+}^n{C}_{2\left[n/2\right]}$, where [ ] denotes greatest integer

• A. $2^{2n-1}$
• B. $2^{2n-1}-1$
• C. $2^{n-1}$
• D. $2^{n-1}-1$

Answer 1

The correct option is C $2^{n-1}$

The real sol. is taking a binomial = $(1+x{)}^n{=}^n{C}_0{+}^n{C}_{1}x{+}^n{C}_2{x}^2+......$
and now taking $(1-x{)}^n{=}^n{C}_0{-}^n{C}_{1}x{+}^n{C}_2{x}^2-......$
Now adding both and putting $x=1$, we get $2^n=2{(}^n{C}_0{+}^n{C}_2{+}^n{C}_4+....)$
=> $2^{n-1}$ = required expression