Qs:
(a) A multiple-choice test contains 10 questions. There are four possible answers to each question. How many ways can a student answer the questions on the test if the student answers every question?
(b) In the scenario of (a): How many ways can a student answer the questions on the test if the student can leave answers blank?
A:
(a) There are 10 questions, so I draw out 10 slots: _ _ _ _ _ _ _ _ _ _. How many choices are there for the first slot? There are 4 possible test answers, so the student has 4 choices:
4 _ _ _ _ _ _ _ _ _ _
How many choices are there for the second slot? Same as above, right? 4 possible answers to the test question:
4 4 _ _ _ _ _ _ _ _
Notice that the student always has 4 choices, so this will result in:
4 4 4 4 4 4 4 4 4 4
So, there are 4*4*4*4*4*4*4*4*4*4 = 410 total possibilities.
(b) This part is very similar, although a student can leave a question blank. With 4 possible answers + blank, that means there are “5 possible ways to answer each question”. Same as above, filling in the 10 slots with 5s:
5*5*5*5*5*5*5*5*5*5 = 510 total possibilities.