Remember, $10000 = 10 ^ {4}$

$2007 \times 20062006 - 2006 \times 20072007$

$= 2007 \times (20060000 + 2006) - 2006 \times (20070000 + 2007)$

$= 2007 \times 20060000 + 2007 \times 2006 - 2006 \times 20070000 - 2006 \times 2007$

$= 1000(2007 \times 2006 - 2006 \times 2007)$

$= 1000(0) = \boxed {0}$

2007 * 2006 * 10001 - 2006 * 2007 * 10001, which by the commutativity of multiplication, the two terms are each other's negatives, so the answer is 0.

Factorizing, it becomes 2007 * 2006 * 10001 - 2006 * 2007 * 10001, which by the commutativity of multiplication, the two terms are each other's negatives, so the answer is 0.

Umm, i don't believe this is the correct way, but here's how i solved it $({ 2006 }^{ 2 }\quad *\quad 2007)\quad -\quad (2006\quad *\quad { 2007 }^{ 2 })$ $2006\quad *\quad 2007\quad -\quad 2006\quad *\quad 2007 = 0$