Compare the numbers:
$P = 888...888 \times  333 .. 333$  ($2010$ digits of $8$ and $2010$ digits of $3$) and  
$Q = 444...444\times  666...6667$  ($2010$ digits of $4$ and $2009$ digits of $6$)

(A): $P = Q$, (B): $P > Q$, (C): $P < Q$.
1. Define \( a \) as a number consisting of 2010 digits of 1:
   \[
   a = \underbrace{111\cdots111}_{2010 \text{ digits of } 1}
   \]

2. Express \( P \) in terms of \( a \):
   \[
   P = 888\cdots888 \times 333\cdots333
   \]
   Notice that \( 888\cdots888 \) (2010 digits of 8) can be written as \( 8a \) and \( 333\cdots333 \) (2010 digits of 3) can be written as \( 3a \). Therefore,
   \[
   P = (8a) \times (3a) = 24a^2
   \]

3. Express \( Q \) in terms of \( a \):
   \[
   Q = 444\cdots444 \times 666\cdots6667
   \]
   Notice that \( 444\cdots444 \) (2010 digits of 4) can be written as \( 4a \) and \( 666\cdots6667 \) (2009 digits of 6 followed by a 7) can be written as \( 6a + 1 \). Therefore,
   \[
   Q = (4a) \times (6a + 1) = 24a^2 + 4a
   \]

4. Compare \( P \) and \( Q \):
   \[
   P = 24a^2
   \]
   \[
   Q = 24a^2 + 4a
   \]
   Clearly, \( Q \) has an additional term \( 4a \) compared to \( P \). Since \( a \) is a positive number (consisting of 2010 digits of 1), \( 4a \) is also positive. Therefore,
   \[
   Q > P
   \]

Conclusion:
\[
\boxed{Q > P}
\]