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NDA2021MathematicsBinomial TheoremActual

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

Options

  1. A2+2^2+2^3+ +2^n
  2. B1+2+2^2+2^3+ +2^n
  3. C1+2+2^2+2^3+ +2^ n-1
  4. D2+2^2+2^3+ +2^ n-1

Correct answer

C. 1+2+2^2+2^3+ +2^ n-1

Step-by-step solution

aligned & (1+x)^n= ^n C₀+ ^n C₁ x+ ^n C₂ x^2+ . .+ ^n C_n x^n & Putting x=1 & (1+1)^n= ^n C₀+ ^n C₁+ ^n C₂+ + ^n C_n aligned 2^n-1= ^n C₁+ ^n C₂+ . .+ ^n C_n ( ^n C₀=1 ) Now, 2^n-1=1+2+2^2+ +2^ n-1 C(n, r)=1+2+2^2+ . .+2^ n-1

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