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MHT CET20255 May 2025Evening ShiftMathematicsBinomial TheoremActual

The value of ⁴⁷ C₄+ _ j=1 ^5 ^ (52-j) C₃ is

Options

  1. A⁵² C ₄
  2. B⁵² C ₂
  3. C⁴⁸ C ₄
  4. D⁴⁸ C ₂

Correct answer

A. ⁵² C ₄

Step-by-step solution

The expression ⁴⁷C₄ + _ j=1 ^5 ^ (52-j) C₃ expands to ⁴⁷C₄ + ⁵¹C₃ + ⁵⁰C₃ + ⁴⁹C₃ + ⁴⁸C₃ + ⁴⁷C₃ . Applying Pascal's identity ^ n C_r + n r-1 = ^ n+1 C_r repeatedly: Combining ⁴⁷C₄ + ⁴⁷C₃ yields ⁴⁸C₄ . Adding ⁴⁸C₃ gives ⁴⁸C₄ + ⁴⁸C₃ = ⁴⁹C₄ . Continuing this pattern through ⁵¹C₃ produces ⁵²C₄ . The expression simplifies to ⁵²C₄ , corresponding to option A.

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