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AP EAMCET20228 Jul 2022Evening ShiftMathematicsBinomial TheoremActual

If ^ n-3 C_r+B^ n-3 C_ r-1 +B^ n-3 C_ r-2 + n-3 r-3 = ^n C_r holds. for all n r 3 , then (B B^ )= .

Options

  1. A(1,5)
  2. B(5,1)
  3. C(3,3)
  4. D(4,2)

Correct answer

C. (3,3)

Step-by-step solution

^ n-3 C_r+B^ n-3 C_ r-1 +B^ n-3 C_ r-2 + n-3 r-3 = ^n C_r ( ^ n-3 C_r+ n-3 r-1 )+ ( n-3 r-1 + n-3 r-2 )+ ( n-3 r-2 + n-3 r-3 )+(B-2) n-3 r-1 + (B^ -2 ) n-3 r-2 = ^n C₃ [ ^ n-2 C_r+ n-2 r-1 ]+ n-2 r-2 + ( n-3 r-1 + n-3 r-2 )+(B-3) n-3 r-1 + (B^ -3 ) n-3 r-2 = ^n C_r ^ n-1 C_r+ [ n-2 r-2 + n-2 r-1 ]+(B-3)^ n-3 C_ r-2 + (B^ -3 ) n-3 r-2 = ^n C_r aligned & ^n C_r+(B-3) n-3 r-1 + (B^ -3 )^ n-3 C_ r-2 = ^n C_r & (B-3) n-3 r-1 + (B^ -3 ) n-3 r-2 =0 aligned It is only possible when, B-3=0 and B^ -3=0 (B, B^ ) (3,3)

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