Question

${}^n{C}_1$ + ${}^n{C}_3$ + ${}^n{C}_5$..............=

• A. 0
• B. 1
• C.
• D.

Answer 1

The correct option is D

$(1+x{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$x + ${}^n{C}_2$$x^2$ + ${}^n{C}_3$$x^3$ + ${}^n{C}_4$$x^4$..........

put x = 1

⇒ $2^n$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$ + ${}^n{C}_3$ + ${}^n{C}_4$ + ${}^n{C}_5$....................(1)

Put x = -1

⇒ 0 = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ - ${}^n{C}_3$ + ${}^n{C}_4$ - ${}^n{C}_5$..............(2)

(1) - (2) ⇒ $2^n$ = 2[${}^n{C}_1$ + ${}^n{C}_3$ + ${}^n{C}_5$..............]

⇒ ${}^n{C}_0$ + ${}^n{C}_0$ + ${}^n{C}_0$......................= $\frac{2^n}{2}$

= $2^{n-1}$