Counting Principles, Combinations and Permutations
By Indrani Banerjee
(Test Preparation (ACT/SAT/SSAT) tutor at The Edge Learning Center)
What are ‘counting principles’? The fundamentals of counting principles appear in some form or another in the IB, AP and A-Levels maths curricula, and are common topics tested in the maths section of the ACT and SAT, so make sure you are prepared to deal with them!
Counting Principles
Think about a time when you went to Ocean Park with friends, and took a ride on the Mine Train. Maybe you had to figure out which order your friends should all queue up in order to sit with the people they wanted. For this challenge, you would need counting principles! It could even be used to figure out the different flavors of ice-cream you’d use to make the best sundaes or cocktails afterward.
(Photo: portlandsonic.com)
Whenever we’re considering the different number of ways something can occur, or if we’re trying to arrange items in a particular order, we are applying the fundamentals of counting principles. These topics are also the two main question types that appear in almost all maths courses, and solving questions on this topic is an important process.
With these steps, it can be relatively straightforward:
- Counting Principle is the method by which we calculate the total number of different ways a series of events can occur. This is always the product of the number of different options at each stage.
Let’s look at an example of this to see how best to apply this principle:
(from ACT 65D, April 2008 paper)
This is a common example of a question that appeared in an actual ACT paper in 2008. In this case, there are 4 events that will occur, and in order to solve this question we need to:
- First, calculate how many different ways each of the four event can occur
| Event | Numbers of Options |
|---|---|
| Choose a sandwich | 4 |
| Choose a soup | 2 |
| Choose a salad | 2 |
| Choose a drink | 2 |
- Then, we can calculate the total number of possible outcomes by multiplying the number of options at each stage.
Total possible outcomes = product of how many different way each selection can be made
Therefore, total number of ways these selections can be made is 4 x 2 x 2 x 2 = 32 possible ways.
Hence, the correct answer is K.
Now, let’s consider the second type of question, where we are asked to consider events where a series of specific objects are drawn from a much larger pool.
2. Let us consider a class of 20 people, out of which we are interested in appointing 4 people for positions of responsibility in the school debate team. Assuming all members of the group are of equal competence and are all capable of carrying out the requirements of the positions effectively; we will need one treasurer, one secretary, one president and finally one vice-president. How many ways can this be done?
We can approach this question in a similar method to the previous question.
- First, let’s work out how many different ways we could pick a person for each position. It is important to note here that whether we pick the treasurer first or the president first will not actually affect the final answer:
| Job title | Number of Options |
|---|---|
| Treasurer | 20 |
| Secretary | 19 (as we have already picked 1 person to be the treasurer) |
| Vice President | 18 |
| President | 17 |
- Therefore, the total number of possibilities for assigning 4 people out of 20 to these positions of responsibility is 20 x 19 x 18 x 17 = 116,280
These topics are sometimes described using the notation nPr and nCr, meaning ‘Permutation’ and ‘Combination’ respectively. For those of us having to survive IB Maths, combinations and permutations (only for HL) come up in both non-calculator and calculator papers. So, let’s have a look at how counting principles fits into the topic of permutations and combinations.
Permutations and Combinations
Permutations and combinations are the various different possible ways we can arrange or select an item or r items out of a sample size of n. You can think about these using our lovely Sets and Venn diagram terminology. If you have a set of n elements and you pick r elements to form a subset, the possible options for this subset are the ‘combinations’. With combinations, the order in which the elements are chosen does not matter, so ABC = CBA =BCA etc. If permutations are being considered, then the order of the elements does create different options, so ABC does not equal CBA etc.
Permutations 
[order matters]
Combinations 
[order doesn’t matter]
Your scientific calculator will always be able to calculate nPr and nCr for you automatically, but here we’ve shown you how they are actually calculated, for people who are curious!
So let’s think about the previous Debate Team question using these two ideas.
- To begin with, we have to ask ourselves if we need to permute or combine for this question? If a person is picked to be the Treasurer, is this the same as being picked or the President? Definitely not! So where the selection of a person is made in the process will make a big difference. If the order in which people are assigned to a role is important, this will be a permutation question!
Here, we can substitute 20 = n and 4 = r values for nPr. Then we have two ways we can solve this. We can either:
- Use a calculator and plug in
Or, if we are feeling particularly assiduous we can always work it out manually by using the formula (given in the IB Data Booklet).
Notice how 16! cancels out from both the numerator and the denominator.
One great application of this is Mr. Potato Head. If you forget which is which, I like to use Potato Head to help me remember. The pieces you choose to put on are your combination of pieces, (the items you have used). The order in which you put them on is you permutation (how you have used them). Therefore, you are combining the parts to make one permutation!
In the ACT there will be no requirement for calculating permutations and combinations manually, as we always have our trusty GDC or scientific calculator which will come to our rescue. In the SAT they usually reserve combinations and permutations for the calculator section of the maths test. The IB gods, however, can ask students to manually calculate combinations in both the non-calculator and calculator papers at SL, and both combinations and permutations can appear in HL. Fortunately, they do provide both of these equations in the data booklets.
Here you can practice these questions for some fun scenarios.
On a final note, remember that whilst this can be a pain to study for a Maths exam, and we can only make so many sundaes before we become fat… we can use these ideas to think about any series of events, from predicting our opponents’ hand of cards in games like poker, through to how best to seat people at a table for a party!
Good luck permutatin’ out there, maths-fans!
Related Blog: Approach to SAT Math
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