Q154 / 270 · NDA 2021 (Phase 1) · Mathematics

If $C_0, C_1, C_2, \ldots, C_n$ are the coefficients in the expansion of $(1+x)^n$, then what is the value of $C_1+C_2+C_3+\ldots+$ $C_n$ ?

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

Solution

$(1+x)^n=C_0+C_1 x+C_2 x^2+C_3 x^3+\ldots .+C_n x^n$ For, $x=1$, $\begin{aligned} & (1+1)^n=C_0+C_1+C_2+C_3+\ldots .+C_n \\ & \Rightarrow 2^n=1+C_1+C_2+C_3+\ldots .+C_n\end{aligned}$ $\left\{\because{ }^n C_0={ }^n C_n=1\right\}$ $\therefore C_1+C_2+C_3+\ldots . .+C_n=2^n-1$

Hence, the correct answer is option: 'B'.

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