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Manipal MET2013MathematicsBinomial Theorem

_ k=0 ¹⁰ ²⁰ C _k is equal to

Options

  1. A2¹⁹+ 1 2 ²⁰ C ₁₀
  2. B2¹⁹
  3. C²⁰ C ₁₀
  4. DNone of these

Correct answer

A. 2¹⁹+ 1 2 ²⁰ C ₁₀

Step-by-step solution

_ k=0 ¹⁰ ²⁰ C _k= ²⁰ C ₀+ ²⁰ C ₁+ + ²⁰ C ₁₀= 1 2 [2 ²⁰ C ₀+2 ²⁰ C ₁+ +2 ²⁰ C ₁₀ ] aligned = 1 2 [ ( ²⁰ C ₀+ ²⁰ C ₂₀ )+ ( ²⁰ C ₁+ ²⁰ C ₁₉ )+ . . +2 ²⁰ C ₁₀ ] aligned = 1 2 [ ( ²⁰ C ₀+ ²⁰ C ₁+ ²⁰ C ₂+ + ²⁰ C ₁₉ . .+ ²⁰ C ₂₀ )+ ²⁰ C ₁₀ ] aligned & = 1 2 [2²⁹+ ²⁰ C ₁₀ ] & =2¹⁹+ 1 2 ²⁰ C ₁₀ aligned

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