Yes, you can enter both lotteries. We didn't have a strong preference, so we entered DC in both lotteries. He got a low number in one (got in) and a really high number in the other program (the entire entering class would have had to refuse their slots before he got in). I'd say entering both lotteries improved DC's chances of getting into an immersion program.

Or do you have the chance to enter DC's name in for both if you don't have a strong preference?