Q196 / 270 · NDA 2021 (Phase 1) · Mathematics

What is $C(n, 1)+C(n, 2)+\ldots+C(n, n)$ equal to?

  • A
    $2+2^2+2^3+\ldots+2^n$
  • B
    $1+2+2^2+2^3+\ldots+2^n$
  • Correct answer
    C
    $1+2+2^2+2^3+\ldots+2^{n-1}$
  • D
    $2+2^2+2^3+\ldots+2^{n-1}$

Solution

$\begin{aligned} & (1+x)^n={ }^n C_0+{ }^n C_1 x+{ }^n C_2 x^2+\ldots . .+{ }^n C_n x^n \\ & \text {Putting } x=1 \\ & (1+1)^n={ }^n C_0+{ }^n C_1+{ }^n C_2+\ldots \ldots+{ }^n C_n\end{aligned}$ $\Rightarrow 2^n-1={ }^n C_1+{ }^n C_2+\ldots . .+{ }^n C_n \quad\left(\because{ }^n C_0=1\right)$ Now, $2^n-1=1+2+2^2+\ldots \ldots+2^{n-1}$ $\therefore \Sigma C(n, r)=1+2+2^2+\ldots \ldots . .+2^{n-1}$

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

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