Problems that involve counting different ways you can order the questions on a true / false quiz can often cause confusion. This is due to the fact that in some problems the different orderings may need to be counted and in other cases the different orders may be irrelevant.
Let’s take a look at some of these types of problems.
Suppose you are taking a true-false quiz with three questions. How many ways are there to answer this quiz?
On the surface, questions like this seem fairly easy. With only three questions, it is fairly easy to make a tree diagram.
As you can see, there are eight ways to reorder three letters selected from T and F. It is not unlike counting the number of ways three coins can be flipped.
Some students might attack the problem using a slot diagram.
Using the Fundamental Counting Principal, we multiply the numbers in the slots to get 2 ∙ 2 ∙ 2 = 8. At a basic level, we are counting the number of ways to order T and F in a three letter word.
TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF
Each ordering of the letters is considered to be different…in other words, order makes a difference. The various ordering are called permutations when order is important.
Now let’s ask a slightly different question. How many ways is there to miss one question on a three question true / false quiz?
In light of this question, let’s change the tree diagram slightly.
Looking at the outcomes in red on the right side, we see there are three ways to get one wrong. Is it possible to get this same answer using the Fundamental Principal?
Now we need to think in terms of tasks or decisions. If we correspond the tasks to the questions, we run into problems. For instance,
This indicates that there is one way to choose the first and second question correctly and the third question wrong…in exactly this order. However, there are other orders that lead to the same outcome, 1 wrong. This slot diagram does not take this into consideration. The reason a strategy like this does not work is that counting how many ways to get one wrong is a different type of problem. When we ask how many ways to get one wrong, we are implying one wrong and two correct. Since we do not distinguish between getting Question 2 and Question 3 correct versus Question 3 and Question 2 correct, order does not make a different. When we count objects without regard to order, the different arrangements are called combinations.
To handle this new question with a slot diagram we need to choose which question to miss. How about a slot diagram with a single slot.
Since we can miss Q1 or Q2 or Q3, there are three ways to do this.
Now let’s complicate it a bit. How many ways is there to get two questions wrong? If we use a slot diagram we might try this:
This seems natural, but is incorrect…let’s list out the number of ways to get two wrong by listing the possibilities.
(Q1, Q2), (Q1, Q3), (Q2, Q3)
(Q2, Q1), (Q3, Q1), (Q2, Q3)
So what is wrong with this? Examine the first entry in each row. These are the exact same questions missed, but in a different order. If we are interested in getting two wrong, these are exactly the same thing. The same goes for the other entries in each row. By reordering, we don’t get a new possibility. So if we use a slot diagram we need to realized that we need to cut the number of possibilities in half since there is two ways to reorder any two questions. The correct number of ways to get two wrong is
Now let’s think about how this would work on a four question quiz where we want to know how many ways is there to get three wrong? Start with a slot diagram and figure out the number of ways to choose a question to miss.
But doing this allows the ways to reorder three questions to be counted multiple times. To count out how many ways there is to rearrange 3 questions, do another slot diagram.
So there are 3 ∙2 ∙ 1 = 6 ways to reorder three questions. So the number of way to get three wrong on a four question T/F quiz is