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COMEDK2018MathematicsBinomial Theorem

If x occurs is the expansion of (x+ 1 x )^ n , then its coefficient is

Options

  1. An ! (r !)¹
  2. Bn ! (r+1) !(r-1) !
  3. Cn ! ( n+r 2 ) ! ( n-r 2 ) !
  4. Dn ! [ ( n 2 ) ! ]²

Correct answer

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

Step-by-step solution

Let x^ r occurs in the expansion of (x+ 1 x )^ n T_ r+1 = n p x^ n-p ( 1 x )^ p n p x^ n-p x^ -p = n p x^ n-2 p let n-2 p=r n-r=2 p p= n-r 2 So, coefficient of x^ r = n p = n ! p !(n-p) ! = n ! ( n-r 2 ) ! (n- n-r 2 ) ! = n ! ( n-r 2 ) ! ( n+r 2 ) !

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