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KVPY2020MathematicsBinomial Theorem

Let M = 2 30 - 2 15 + 1 , and M 2 be expressed in base 2 . The number of 1 ' s in this base 2 representation of M 2 is

Options

  1. A29
  2. B30
  3. C59
  4. D60

Correct answer

B. 30

Step-by-step solution

2 n 2 = 1 00 … 0 ⏟ n  times  M 2 = 2 60 - 2 46 + 2 30 - 2 16 + 2 31 + 1 = ( 11 … 1 ⏟ 14  times  00 … 0 ⏟ 46  times  + 11 … 1 ⏟ 14  times  000 … 0 ⏟ 16  times  + 1 00 … 0 ⏟ 31  times  + 1 ) 2 Number of 1 ' s = 14 + 1 + 14 + 1 = 30

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