Question

The sum of the series ${}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\dots {+}^n{C}_n$ equals

• A. $2^n-1$
• B. $2^n$
• C. $2^n+1$
• D. none of these

Answer 1

The correct option is A

$2^n-1$

Step 1: Combination

${}^{n}C_r$ represents the selection of objects from a group of objects where order of objects does not matter.

${}^n{C}_r=\frac{{\displaystyle n!}}{r!(n-r)!}$

Step 2: Explanation for correct answer

Option (a): $2^n-1$

We know that,

$(1+x{)}^n{=}^n{C}_0{x}^0{+}^n{C}_{1}x{x}^1{+}^n{C}_2{x}^2{+}^n{C}_3{x}^3\dots .{+}^n{C}_n{x}^n$

${}^n{C}_r=\frac{{\displaystyle n!}}{r!(n-r)!}$

Put $x=1$

$\begin{array}{r}(1+1{)}^n{=}^n{C}_0\times 1^0{+}^n{C}_1\times 1^1{+}^n{C}_2\times 1^2{+}^n{C}_3\times 1^3\dots .{+}^n{C}_n\times 1^n\end{array}$

$2^n=1{+}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\dots .{+}^n{C}_n$

${}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\dots {+}^n{C}_n={22}^n-1$

The sum of the series ${}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\dots {+}^n{C}_n$ equals to $2^n-1$.

Thus, option (a) is the correct answer.

Step 3: Explanation for incorrect answer

Option (b): $2^n$

Option (c): $2^n+1$

Option (d): none of these

Therefore, by calculating the sum of the series ${}^n{C}_1{+}^n{C}_2{+}^n{C}_3+\dots {+}^n{C}_n$ we get $2^n-1$.

Thus, option (b), (c) and (d) are incorrect answers.

Hence, option (a) is the correct answer.