We recently wrote a post about teaching permutations and today we refer to the topic of combinations. The challenges that learners face in regard to problems on combinations are, of course, quite similar to those they face with permutations. The logical thinking required often turns out to be a major obstacle in both these areas of the syllabus.

Once a particular problem has been correctly identified as a permutation-type or a combination-type, there will usually still be some distance to go in all but the most basic of questions. Simply making the correct decision to press the * ^{n}C_{r}* button rather than the

*button on the calculator is rarely sufficient to assure a correct answer.*

^{n}P_{r}Again, learners need to develop an appreciation that there is seldom only one route to a correct solution. Also, in dealing with combinations, it is crucial to understand that two or more different arrangements of a set of objects are, in fact, just one single combination of those objects.

In our Cambridge International AS & A Level Mathematics: Probability and Statistics 1 Coursebook, this type of problem is illustrated in Worked example 5.14 and learners will meet it in several of the questions in Exercise 5F, End-of-Chapter Review Exercise 5 and Cross-topic Review Exercise 2.

The most appropriate point at which to offer this presentation to your learners would be after attempting Question 13 in Exercise 5F, which is where they will first need to avoid using one of the methods featured in the presentation.

The presentation looks at three different approaches that might be used to solve one particular combination problem. The problem concerns combinations in which at least one of each of two types of object must be included. Only two of the three given approaches are successful. There is a brief discussion on this and, needless to say, learners will benefit greatly if they can appreciate the reason why one of the approaches fails.

Four extension activities with solutions are also offered at the end of the presentation to stretch your students.

To view the PowerPoint, click here.

**About the author**

Dean Chalmers is an experienced author and teacher having previously taught mathematics in the UK, Vietnam, Malaysia and Botswana. Dean is the author of our Cambridge International AS & A Level Probability & Statistics 1 and Cambridge O Level Statistics coursebooks and has also contributed to our UK AS/A Level Further Mathematics Statistics coursebooks.