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COMEDK2023Morning ShiftMathematicsBinomial TheoremActual

In the expansion of (1+3 x+3 x^2+x^3 )^ 2 n , the term which has greatest binomial coefficient, is

Options

  1. A(3 n-1) th term
  2. B(3 n+1) th term
  3. C(3 n+2) th term
  4. D(3 n) th term

Correct answer

B. (3 n+1) th term

Step-by-step solution

The given expression is (1+3x+3x^2+x^3 )^ 2n . Recognizing the binomial expansion, 1+3x+3x^2+x^3 = (1+x)^3 . Substituting this into the expression, we get ((1+x)^3)^ 2n = (1+x)^ 6n . The expansion of (1+x)^ 6n is given by _ r=0 ^ 6n ^ 6n C_ r x^r . The binomial coefficients are of the form ^ 6n C_ r . The greatest binomial coefficient in the expansion of (1+x)^N occurs at the middle term. For N=6n , the middle term is at r = 6n 2 = 3n . The general term T_ r+1 in the expansion of (1+x)^ 6n is ^ 6n C_ r x^r . For r=

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