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Most Important Selected Qs for JEE AdvancedMathematicsBinomial Theorem

If (1+x)^n=C₀+C₁ x+C₂ x^2+ +C_n x^n , then the sum of the product of the coefficients taken two at a time can be represented by _ i=0 ^n _ j=i+1 ^n C_i C_j=2^a- b ! c(d !)^2 . Then which of the following are correct?

Options

  1. Aa=2 n-1
  2. Bb=2 n
  3. Cc=2
  4. Dd=n

Correct answer

D. d=n

Step-by-step solution

Note that C_k is binomial coefficient ^n C_k . ( _ i=0 ^n C_i )^2= _ i=0 ^n C_i^2+2 _ i=0 ^n _ j=i+1 ^n C_i C_j _ i=0 ^n _ j=i+1 ^n C_i C_j= 1 2 [ ( _ i=0 ^n C_i )^2- _ i=0 ^n C_i^2 ]= 1 2 [ ( _ i=0 ^n ^n C_i )^2- _ i=0 ^n ( ^n C_i )^2 ]= 1 2 [2^ 2 n - ^ 2 n C_n ]=2^ 2 n-1 - (2 n) ! 2(n !)^2 Therefore, a=2 n-1, b=2 n, c=2, d=n all are true.

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