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Manipal MET2011MathematicsBinomial Theorem

If x+y=1 , then _ r=0 ^n r^ 2 n C_r x^r y^ n-r is equal to

Options

  1. An x y
  2. Bn x(x+y n)
  3. Cn x(n x+y)
  4. DNone of these

Correct answer

C. n x(n x+y)

Step-by-step solution

aligned _ r=0 ^n & r^2 ^n C_r x^r y^ n-r & = _ r=0 ^n[r(r-1)+r] ^n C_r x^r y^ n-r & = _ r=0 ^n r(r-1)^n C_r x^r y^ n-r + _ r=0 ^n r^n C_r x^r y^ n-r aligned aligned = & _ r=2 ^ n-2 r(r-1) n r n-1 r-1 n-2 r-2 x^2 x^ r-2 y^ n-r + _ r=1 ^n r n r n-1 r-1 x x^ r-1 y^ n-r aligned =n(n-1) x^2 _ r=2 ^ n-2 n-2 r-2 x^ r-2 y^ (n-2)-(r-2) +n x _ r=1 ^n n-1 r-1 x^ r-1 y^ (n-1)-(r-1) =n(n-1) x^2+n x ( x+y=1)=n x(n x-x+1)=n x(n x+y) ( x+y=1)

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