How to find the fourth term of a binomial expansion?

So let's look at how to expand a binomial ...

Start with just the variable exponents which will be numbered in ascending and descending order.

example (without coefficients)

(x+y) ^3 = x^3 (y^0) + x^2 (y^1) + x^1 (y^2) + x^0 (y^3)

If the binomial is (x-y) ^3 the + / - signs will toggle like so ... (again leaving out coefficients)

(x-y) ^3 = x^3 (y^0) - x^2 (y^1) + x^1 (y^2) - x^0 (y^3)

If there is a coefficient or power inside the binomial like (x^2+2y) ^3 these are also raised to the appropriate power.

(x^2+2y) ^3 = (x^2) ^3 (y^0) (2^0) + (x^2) ^2 (y^1) (2^1) + ...

now there are also coefficients for each term which follow a combination pattern of Pascal triangle. C (3,0); C (3,1); C (3,2); C (3,3)

these get multiplied by each term.

So the 4th term of (x^2-2y) ^3

will be negative

have coefficient of: C (3,3) * (2^3) = 8

variables: x^0 (y^3) = y^3

= - 8y^3