The coefficient of $y^4$

$\begin{aligned} \frac 1{(1-2y)^3} & = (1-2y)^{-3} \\ & = 1 + (-3)(-2y) + \frac {(-3)(-4)}{2!}(-2y)^2 + \frac {(-3)(-4)(-5)}{3!}(-2y)^3 + \frac {(-3)(-4)(-5)(-6)}{4!}(-2y)^4 + \cdots \end{aligned}$

Therefore the coefficient of $y^4$ is $\dfrac {(-3)(-4)(-5)(-6)}{4!}(-2)^4 = \boxed{240}$ .

For any index $n$ ,

$(1+x)^n=1+nx+\frac {n(n-1)}{2!} x^2+\frac {n(n-1)(n-2)}{3!} x^3.....$

Thus, coefficient of $y^4$ would be

$2^4\cdot \frac {-3\cdot -4 \cdot -5\cdot -6}{4!}$

$=16×15=240$

$\frac{1}{(1 - 2y)}$ = 1 + 2y + 4y^2 + 8y^3 + 16y^4 + . . . and $\frac{1}{(1 - 2y)^2}$ = (1 + 2y + 4y^2 + 8y^3 +16y^4 + . . .)^2 = 1 + 4y + 12y^2 + 32y^3 + 80y^4 + . . . . Further $\frac{1}{(1 - 2y)^3}$ = 1 + 6y + 24y^2 + 80y^3 + 240y^4 + . . . .Powers of y greater than 4 can't contribute to the power of y^4 in the result. Thus the coefficient of y^4 in $\frac{1}{(1 - y)^3}$ = 240