The Obama administration chose to publish the ten-year cost number, because that makes them look better than any other choice. Too short and the startup costs aren't amortized much, too long and you get into the time period where we're scheduled to actually pay for much of it. Those 12 million people wil need insurance for the next ten years, so it's perfectly appropriate to talk about what it will cost to cover them for ten years. 12 million isn't a one-time number, as if they only needed coverage for one day. The number of previously uninsured people may covered may fluctuate a bit, but not by order of magnitude or anything like that.
No, that's not a mistake I made. I made a much simpler mistake, though - I lost track of the number of zeros ehile trying to calculate trillions in my head.