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AP EAMCET201922 Apr 2019Morning ShiftMathematicsBinomial TheoremActual

The coefficient of (x^5 ) in the expansion of ( (x^2+2 x+3 )^5 ), is

Options

  1. A1052
  2. B540
  3. C480
  4. D1020

Correct answer

A. 1052

Step-by-step solution

Given, ( (3+2 x+x^2 )^5 ) (= n ! (p !)(q !)(r !) (3)^p(2 x)^q (x^2 )^r )...(i) Where, ( p+q+r=n=5 ) For (x^5 ), we have, (q+2 r=5 ) ( aligned & r=0, q=5, p=0 & r=1, q=3, p=1 & r=2, q=1, p=2 aligned ) Putting the above values in Eq. (i), we have ( aligned & 5 ! 0 ! 5 ! 0 ! (3)^0(2)^5(1)^0+ 5 ! 1 ! 3 ! 1 ! (3)^1 (2^3(1)^1 . & = 5 ! 2 ! 1 ! 2 ! (3)^2(2)^1(1)^2 & =32+20 3 8+30 9 2 & 32+480+540=1052 aligned )

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