Permutation and combination are two ways to quantify various arrangements in which objects from a set may be selected to form subsets. They provide two counting principles to group elements from a larger set into a selective one. In general, this endeavor is done without replacement of the prior objects. Particularly, such selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.
As discussed above, permutations and combinations differ in whether the order of the elements matters.
In order to understand permutation and combination, I had to firstly introduce the concept of factorials to my students. A factorial of a given number n (denoted as n!) is the product of the first n natural numbers. This concept is applicable in counting the number of ways n objects can be arranged without replacement as n! will mathematically reflect such an arrangement outcome.
Let us briefly elaborate on the two distinct cases based on simple cases where it is easy to distinguish between permutation and combination. I intend to do so, in order to show a more straightforward undertaking before discussing an interesting case. I had to teach students in which they seemed a bit bewildered as far as which counting technique to use.
A permutation is an arrangement in a particular order of a number of objects selected at a time. For instance, let us consider the first ten whole numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The number of different 4-digit-PIN that can be formed using these numbers is 5040. P (n,r) where n is the total object and r is the selected will provide us with the result : P(10,4) = 5040. This is a simple example where we applied the idea to use the factorial of 10 divided by the factorial of the difference of 10 and 4 from the formula (which can be enunciated as “n choose r”) nPr = n!/(n−r)!
A combination, as compared to permutation, emphasizes more on the idea of ‘grouping’ than ‘arrangement’. For example, let us take the case of building a swimming team to compete in a tournament in which 2 students are to be selected from a total of 5 students. This combination of ‘r’ team members to be taken from the available ‘n’ total members is given as nCr=n!/(r!*(n−r)!). The combination, where it does not matter which student was selected at first, could be formed (as per application of this rule) in a total of 10 ways. I took this point to elaborate to my students that the only difference between two counting principles is the presence of an extra ‘r’ factorial in the denominator of the combination formula. This should make sense, as here, we are trying to reduce the number of outcomes possible, if we know that order does not matter. Alternatively, a permutation case would have given us more possible selected arrangements because the order creates more possibilities, and as such, it comparatively provides a greater magnitude of result.
Having elaborated on the basic concepts of both the techniques, I will now share an interesting experience of teaching them in class. Particularly, I had provided an assignment to my students to identify whether a certain case is a permutation or combination. Please refer to image A.
Specifically, my students had a problem on question 10 and 11 (in image B) that intrigued me while I provided my subsequent clarification of its solution.
Interestingly, one of the students did not seem to understand why the solution for question 10 in image B entailed a permutation instead of combination. Indeed, there seems to be a nuanced distinction that I clarify for the student for this problem. There are three positions here, namely left wing, center and right wing that are in ‘in order’ even though they are not explicitly in order such as in case of first, second and third winning titles at a competition. Hence , this is also case of permutation as ‘positions’ are similar to ‘titles’, as the order matters in both.