While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards.  Let $a$ be the probability that the cards are of different ranks.  Compute $\lfloor 1000a\rfloor$.
To solve this problem, we need to calculate the probability that two cards drawn from a standard deck of 52 playing cards have different ranks.

1. **Calculate the total number of ways to draw two cards from the deck:**
   \[
   \binom{52}{2} = \frac{52 \times 51}{2} = 1326
   \]

2. **Calculate the number of ways to draw two cards of the same rank:**
   There are 13 ranks in a deck, and for each rank, there are \(\binom{4}{2}\) ways to choose 2 cards of that rank.
   \[
   \binom{4}{2} = \frac{4 \times 3}{2} = 6
   \]
   Therefore, the total number of ways to draw two cards of the same rank is:
   \[
   13 \times 6 = 78
   \]

3. **Calculate the number of ways to draw two cards of different ranks:**
   Subtract the number of ways to draw two cards of the same rank from the total number of ways to draw two cards:
   \[
   1326 - 78 = 1248
   \]

4. **Calculate the probability that the two cards have different ranks:**
   \[
   a = \frac{1248}{1326}
   \]

5. **Simplify the fraction:**
   \[
   a = \frac{1248}{1326} = \frac{208}{221}
   \]

6. **Compute \(\lfloor 1000a \rfloor\):**
   \[
   1000a = 1000 \times \frac{208}{221} \approx 941.176
   \]
   Therefore,
   \[
   \lfloor 1000a \rfloor = 941
   \]

The final answer is \(\boxed{941}\)