Binomial expansion

  • Thread starter sara_87
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  • #1
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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?.
 

Answers and Replies

  • #2
EnumaElish
Science Advisor
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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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