Determine the coefficient of y^3 in the binomial expansion (2x-3y)^4

Using the method of binomial expansion (which I will cover in more detail) we get

(2x-3y)^4 = + 1(2x)^4(3y)^0  -  4(2x)^3(3y)^1  +  6(2x)^2(3y)^2  -  4(2x)^1(3y)^3  +  1(2x)^0(3y)^4  =

= 16x^4  -  96x^3 y  +  216x^2 y^2  -  216x y^3  +  81y^4

Note that we can get the coefficients 1, 4, 6, 4, 1  from Pascal's triangle, and since in the given example there is subtraction (2x-3y), there is a minus sign before each term that has 3y in an odd factor (^1, ^3 etc). You can simply remember to add a minus sign before every second term.

Now we see that in the term where y is in factor 3 as asked in the question (this is the term -216xy^3), the coefficient is -216. This is the answer we are looking for!