Question

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

two choices for each answer, six answers, by the counting principle number of possibitlities is
\[2\times 2\times 2\times 2\times 2\times 2=2^6=64\]

Answer 2

if you can leave some blank then there are 3 possibilities for each answer
adjust accordingly

Answer 3

thank you