Methods to find coefficient of a term in sum of polynomials

Question

I have a polynomial sequence that I would like to sum, that goes as follows:

$1 – (y-1) + (y-1)^2 – (y-1)^3 + … + (y – 1)^{17}$

which is basically

$\sum^{17}_{r = 0} (-1)^r (y-1)^r$

I would like to find the coefficient of coefficient of $y^4$ in the expansion of this summation.

Right now the only method I can think of to solve this is to one by one use binomial expansion to expand each term, and adding up the values of each term to get the final coefficient of $y^4$ however I believe there is a shorter way to get to the solution.

Does anyone have any way or method to get the solution to this problem in a shorter way?

Answer 1

Hint: this sum can be rewritten as

$$\frac{1-{(y-1)}^{18}}{y}$$ (you can read https://en.wikipedia.org/wiki/Geometric_progression)

or coefficient of $y^4$ is $\binom{18}{5}$