An exam consists of six true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered?

Answer 1

Since there are 2 choices for each answer, there are 2^6 = 64 ways to answer all the questions.

If some may be left unanswered, then we have

2^6 * 6C6 ways to select and answer all 6
2^5 * 6C5 ways to select and answer 5
2^4 * 6C4 ways to select and answer 4
and so on. Adding them all up, you get 728