Question

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

Answer 1

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

Explanation:

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...