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AP EAMCET202315 May 2023Evening ShiftMathematicsBinomial TheoremActual

If C_j= ^n C_j , then C₀ C_r+C₁ C_ r+1 +C₂ C_ r+2 + +C_ n-r C_n=

Options

  1. A(2 n) ! (n-2 r) !(n+2 r) !
  2. B(2 n) ! (n-r) !(n+r) !
  3. C2 n _ C _ r
  4. D2 n_ C_ r+1

Correct answer

B. (2 n) ! (n-r) !(n+r) !

Step-by-step solution

aligned & Since (1+x)^ 2 n =(1+x)^n(x+1)^n & (1+x)^ 2 n = (C₀+C₁ x+C₂ x^2+ +C_n x^n ) & (C₀ x^n+C₁ x^ n-1 +C₂ x^ n-2 + +C_n ) aligned Now equation co-efficient of x^ n-r on both side C₀ C_r+C₁ C_ r+1 +C₂ C_ r+2 + +C_ n-r C_n= 2 n n+r

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