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Binomial expansion

  1. Nov 14, 2007 #1
    Apply the binomial expansion to : (1+x)^n and show that the coefiicient of x^n in the expansion of (1+x)^2n is:
    (nC0)^2 +(nC1)^2 +...+(nCn)^2
    hint: (nCm)=(nC(n-m))

    my approach:

    (1+x)^n = x^n + nx^(n-1) + (nC2)x^(n-2) +...+ 1

    (1+x)^2n = x^(2n) + nx^(2n-1) +...+ x^n

    i dont know what to do next. it looks easy but i cant figure it out.
    can someone help me please?.
     
  2. jcsd
  3. Nov 14, 2007 #2

    EnumaElish

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    (1+x)^2n = ((1+x)^n)^2 = (x^n + nx^(n-1) + (nC2)x^(n-2) +...+ 1)(x^n + nx^(n-1) + (nC2)x^(n-2) +...+ 1).

    Now you need to cross-multiply and verify which cross-multiplied terms simplify to x^n. For ex., (Ax)Bx^(n-1) = (AB)x^n.
     
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