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Manipal MET2021MathematicsBinomial Theorem

In the usual notation, ^n C₁ 2 + ^n C₂ 3 + + ^n C_n n+1 is equal to

Options

  1. A2^ n+1 -1 n+1
  2. B2^ n+1 -n-1 n+1
  3. C2^ n+1 -n n+1
  4. DNone of these

Correct answer

D. None of these

Step-by-step solution

aligned & ^n C₁ 2 + ^n C₂ 3 + + ^n C_n n+1 = _ r=0 ^n ^n C_r r+1 - ^n C₀ = & _ r=0 ^n 1 n+1 n+1 r+1 ^n C_r- ^n C₀ = & 1 n+1 _ r=0 ^n n+1 r+1 - ^n C₀ aligned aligned & = 1 n+1 [ ( ^ n+1 C₀+ ^ n+1 C₁+ + n+1 n+1 )- ^ n+1 C₀ ]-C₀ & = 1 n+1 (2^ n+1 -1 )-1 & = 2^ n+1 -n-1-1 n+1 = 2^ n+1 -n-2 n+1 aligned

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