Binomial Expansion of (1+x)^n: Coefficient of x^n

[sara_87]

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 don't know what to do next. it looks easy but i can't figure it out.
can someone help me please?.

 

[EnumaElish]

(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.