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NDA2022MathematicsBinomial TheoremActual

What is the value of C(51,21)-C(51,22)+C(51,23)-C(51,24)+C(51,25)-C(51,26)+C(51,27)-C(51,28)+C(51,29)-C(51,30) ?

Options

  1. AC(51,25)
  2. BC(51,27)
  3. CC(51,51)-C(51,0)
  4. DC(51,25)-C(51,27)

Correct answer

C. C(51,51)-C(51,0)

Step-by-step solution

aligned & 51 21 - 51 22 + 51 23 - 51 24 + 51 25 - 51 26 + 51 27 - 51 28 & + 51 29 - ⁵¹ C³⁰ & = 51 51-21 - 51 51-22 + 51 51-23 - 51 51-24 + 51 51-25 - & 51 26 + 51 27 - 51 28 + 51 29 - 51 30 [ ^ n C_ r = ^ n C_ n -1 ] & = 51 30 - 51 29 - 51 28 - 51 27 + 51 26 - 51 26 + 51 27 - 51 28 & + 51 29 - 51 30 & =0 & Now, 51 51 - ⁵¹ C₀=1-1=0 aligned

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