Question

State true or false.

$C_0+C_1+C_2+\dots +C_n=2^n$

• A. True
• B. False

Answer 1

The correct option is A True

From the binomial expansion

${(1+x)}^n{=}^n{C}_0{+}^n{C}_{1}x{+}^n{C}_2{x}^2+...{+}^n{C}_n{x}^n$ ......$\left(1\right)$

Now substituting $1=1$ on both sides, we have

${(1+1)}^n{=}^n{C}_0{+}^n{C}_{1}1{+}^n{C}_2{1}^2+...{+}^n{C}_n{1}^n$

$2^n{=}^n{C}_0{+}^n{C}_1{+}^n{C}_2+...{+}^n{C}_n$

$\therefore {}^n{C}_0{+}^n{C}_1{+}^n{C}_2+...{+}^n{C}_n=2^n$

Hence the given statement is true.