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MHT CET202523 Apr 2025Evening ShiftMathematicsBinomial TheoremActual

If ¹⁵ C ₄+ ¹⁵ C ₅+ ¹⁶ C ₆+ ¹⁷ C ₇+ ¹⁸ C ₈= ¹⁹ C _ r , then the value of r is equal to

Options

  1. A9 or 10
  2. B7 or 12
  3. C8 or 10
  4. D8 or 11

Correct answer

D. 8 or 11

Step-by-step solution

Given equation: 15 4 + 15 5 + 16 6 + 17 7 + 18 8 = 19 r Apply Pascal’s Identity n r + n r+1 = n+1 r+1 consecutively: 15 4 + 15 5 = 16 5 The series becomes 16 5 + 16 6 + 17 7 + 18 8 16 5 + 16 6 = 17 6 17 6 + 17 7 = 18 7 The expression simplifies to 18 7 + 18 8 = 19 8 Thus 19 8 = 19 r This equality holds when r=8 or r=19-8=11 The solution corresponds to option D .

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