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JEE Main20249 Apr 2024Morning ShiftMathematicsBinomial TheoremActual

The coefficient of x⁷⁰ in x^2(1+x)⁹⁸+x^3(1+x)⁹⁷+x^4(1+x)⁹⁶+ +x⁵⁴(1+x)⁴⁶ is ⁹⁹ C _ p - ⁴⁶ C _ q . Then a possible value of p+q is :

Options

  1. A55
  2. B83
  3. C61
  4. D68

Correct answer

B. 83

Step-by-step solution

aligned & x^2(1+x)⁹⁸+x^3 (1+x⁹⁷ )+x^4(1+x)⁹⁶+ & x⁵⁴(1+x)⁴⁶ aligned Coeff. of x ⁷⁰: ⁹⁸ C ₆₈+ ⁹⁷ C ₆₇+ ⁹⁶ C ₆₆+ aligned & ⁴⁷ C ₁₇+ ⁴⁶ C ₁₆ & = ⁴⁶ C ₃₀+ ⁴⁷ C ₃₀+ . . ⁹⁸ C ₃₀ & = ( ⁴⁶ C ₃₁+ ⁴⁶ C ₃₀ )+ ⁴⁷ C ₃₀+ . ⁹⁸ C ₃₀- ⁴⁶ C ₃₁ & = ⁴⁷ C ₃₁+ ⁴⁷ C ₃₀+ . . ⁹⁸ C ₃₀- ⁴⁶ C ₃₁ & & = ⁹⁹ C ₃₁- ⁴⁶ C ₃₁= ⁹⁹ C _ p - ⁴⁶ C _ q aligned Possible values of (p+q) are 62,83,99,46 p+q=83

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