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