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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2021 · Mathematics · Binomial Theorem

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

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

Options

  1. A2+22+23+…+2n2+2^2+2^3+\ldots+2^n
  2. B1+2+22+23+…+2n1+2+2^2+2^3+\ldots+2^n
  3. C1+2+22+23+…+2n−11+2+2^2+2^3+\ldots+2^{n-1}
  4. D2+22+23+…+2n−12+2^2+2^3+\ldots+2^{n-1}

Correct answer

C. 1+2+22+23+…+2n−11+2+2^2+2^3+\ldots+2^{n-1}

Step-by-step solution

(1+x)n=nC0+nC1x+nC2x2+…..+nCnxnPutting x=1(1+1)n=nC0+nC1+nC2+……+nCn\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{ }^ nC_0=1\right) Now, 2n−1=1+2+22+……+2n−1∴ΣC(n,r)=1+2+22+……..+2n−12^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}
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