Expand (1-2x)6 using binomial theorem

$a = 1$
$b = -2x$
$n = 6$

$C(6, 0) = 1$
$C(6, 1) = 6$
$C(6, 2) = \frac{6 \times 5}{2 \times 1} = 15$
$C(6, 3) = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$
$C(6, 4) = C(6, 2) = 15$
$C(6, 5) = C(6, 1) = 6$
$C(6, 6) = 1$

Term 1 ($r=0$): $C(6, 0) (1)^{6-0} (-2x)^0 = 1 \cdot 1^6 \cdot 1 = 1$
Term 2 ($r=1$): $C(6, 1) (1)^{6-1} (-2x)^1 = 6 \cdot 1^5 \cdot (-2x) = -12x$
Term 3 ($r=2$): $C(6, 2) (1)^{6-2} (-2x)^2 = 15 \cdot 1^4 \cdot (4x^2) = 60x^2$
Term 4 ($r=3$): $C(6, 3) (1)^{6-3} (-2x)^3 = 20 \cdot 1^3 \cdot (-8x^3) = -160x^3$
Term 5 ($r=4$): $C(6, 4) (1)^{6-4} (-2x)^4 = 15 \cdot 1^2 \cdot (16x^4) = 240x^4$
Term 6 ($r=5$): $C(6, 5) (1)^{6-5} (-2x)^5 = 6 \cdot 1^1 \cdot (-32x^5) = -192x^5$
Term 7 ($r=6$): $C(6, 6) (1)^{6-6} (-2x)^6 = 1 \cdot 1^0 \cdot (64x^6) = 64x^6$

The expansion of $(1-2x)^6$ is $\boxed{1 - 12x + 60x^2 - 160x^3 + 240x^4 - 192x^5 + 64x^6}$.