A topic that students generally find quite challenging at AS Level is permutations and combinations. Experience as teachers shows us that a high proportion of candidates make confused attempts to solve all but the most basic of these types of questions.
A reason sometimes given for this by students is ‘We’ve never done this kind of maths before’. It will usually not take your learners long to realise that the mathematics required in this topic is actually quite simple – it is rather the unfamiliar demands of logical thinking and the fact that there is no ‘formula’ for obtaining a correct answer that cause problems for the majority.
Many students spend a large amount of time, and cause themselves a considerable amount of stress, pondering over the question ‘Is it a perm or is it a comb?’ Then, once they have reached a decision, give insufficient thought as to what to do with the numbers that appear in a question. Confidence can be boosted by allowing time for discussing strategies to help learners distinguish one type of question from the other. Knowledge and understanding of key words (arranged for a permutation and select/choose for a combination) is the basis from which such discussions can develop.
Importantly, learners also need to develop an appreciation that there is rarely only one way of solving a permutation or combination problem. A variety of approaches can usually be used to arrive at a correct solution, just as a variety of approaches can be used to arrive at an incorrect solution. In this latter case, it is nearly always the logic, rather than the mathematics, that is flawed.
The PowerPoint presentation below looks at three different approaches that can be taken to successfully solve one particular permutation problem. The problem concerns arrangements in which certain objects must all be separated from each other.
In our Cambridge International AS & A Level Mathematics:Probability & Statistics 1 Coursebook, two efficient methods for dealing with this type of situation appear in Worked example 5.6 on page 130 and in Worked example 5.10 on page 133. A third method features in this presentation, but is not considered in the Worked examples in the coursebook because of its limitations.
However, a little later in the coursebook in the Explore 5.3 activity on page 138, learners are given the opportunity to consider these limitations, and one of the aims of this presentation is to help consolidate an understanding of them. The most effective point to offer this presentation would be after attempting the Explore 5.3 activity or, for a particularly advanced group of learners, prior to studying Section 5.3: Combinations on page 135.
You can download the PowerPoint here.