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From what I remember, it's actually pretty tough to make it on the show.
First Round: People wishing to participate must call an 800 number and correctly answer 3 multiple choice questions. You have to enter your social security number and birthdate, and so you can only play once a day (if you can even get through....the phone lines are always busy). Of the people that correctly answered the questions, they are then placed in a random drawing.
Second Round: From the random drawing, 40 semi-finalists are selected to play in the second round. They are all be called at the same time, and each person must correctly answer five questions that are much harder than the three in the first round. Out of those that get all five correct, 10 are randomly selected to play on TV in the final round.
But if I remember correctly, then rarely does 10 or more people get all five (or even four) questions correct; then they randomly choose from the highest scorers. This is where the idiot factor comes in - it's easy to keep trying in round one until you get all three correct, so with a little luck in round two just about anybody can make it if they try hard enough.