Sorry, Melody. I know this is a slightly hard problem, at least for me. I looked over your working to comprehend it, and it looks a bit like what I had done. My answer had only a slight arithmetic error, and hence the difference.
However, both of our answers appear to be wrong! Here is the answer I recieved from my source:
There are 10 options for the numeric value of the two cards in the pair. Once we selected it, there are C(5,2)=10 ways to pick which two suits the cards in the pair will come from. There are then C(9,3) = 84 ways to choose the numbers on the final three cards, and 5 choices for the suit for each card, giving us a total of 10*10*84*5³ = 1,050,000 ways.