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.