What is coefficient of the #x^4# term in the binomial expansion of # (x^2-1)^12#?

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Answer 1 · 2015-07-16T18:29:19

The coefficient of the #x^4# term is #(-1)^10((12),(10)) = 66#

The binomial expansion of #(x^2-1)^12# is:

#sum_(n=0)^(n=12) ((12),(n)) (-1)^n(x^(24-2n))#

#=((12),(0))x^24-((12),(1))x^22+((12),(2))x^20-...#

#+((12),(10))x^4-((12),(11))x^2+((12),(12))#

So the coefficient of the #x^4# term is:

#((12),(10)) = (12!)/(10!2!) = (12*11)/2 = 66#

Alternatively, write down Pascal's triangle as far as the #13#th row and pick out the coefficient from there...

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