In how many ways one can answer a 20-question exam with two-choice questions? There is a problem in my book:

In how many ways one can answer a 20-question exam with two-choice questions?

which I would solve as follows:
Since there are 20 questions, and each question has two choices, there are $20 \cdot {{2} \choose 1}$ but I am wrong according to the solution written in the book. Why I can't do it this way? I know what the correct answer is, I just want to know why my solution doesn't work in this case.

Think of a three question test.  For each answer on question 1 you have two choices for question 2, for a total of 4.  Now for each of the 4 you have 2 choices for question 3, making 8.  So the choices multiply.
