This module is about counting the number of ways something can be done: arranging people or digits in order (permutations) and choosing a group where order does not matter (combinations). One idea runs through all of it: decide first whether order matters, then decide whether any condition restricts the choices.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content in your exam year, including which counting situations are included and which formulae are given. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need factorials (5! = 5 × 4 × 3 × 2 × 1 = 120), the multiplication principle (a task done in stages, with 3 choices then 4 choices, has 3 × 4 = 12 outcomes) and comfortable fraction cancelling. If cancelling long products by hand is slow, the non-calculator working trainer gives you practice at it.
An orienting example
Eight students are in a club. Find (a) the number of ways to choose 3 of them for a trip, and (b) the number of ways to choose 3 of them and seat them in three labelled seats.
Part (a), order does not matter: a trip group of Aini, Ben and Chong is the same group as Chong, Aini and Ben. So use a combination: 8C3 = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.
Part (b), order matters: each group of 3 can be seated in 3! = 6 ways, so the number is 56 × 6 = 336. This is the permutation 8P3 = 8 × 7 × 6 = 336.
Check: 336 / 6 = 56, so the two answers agree. That link, nPr = nCr × r!, is worth remembering, because counting questions are frequently a selection followed by an arrangement.
In which order should you study it?
- Decide whether order matters: the first decision in every question, and the one that chooses between nPr and nCr.
- Count arrangements with a fixed position: fills the restricted place first, which prevents most errors.
- Count selections with a restriction: handles “exactly”, “at least” and “must include” by splitting into cases.
- Use a complement to simplify counting: subtracts the unwanted outcomes instead of adding up a long list of cases.
- Explain overcounting in a proposed method: tests whether a plausible method counts some outcomes more than once.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Using nPr for a group when the question only asks who is chosen, not who does what.
- Choosing the free positions first and then finding the restricted position has lost its options.
- Adding instead of multiplying when two independent choices are made together, such as 2 girls and 3 boys.
- Missing a case in “at least” questions, or including one twice.
- Trusting a method that sounds logical, such as “choose one special person first”, when it counts some groups more than once.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper and write one line saying whether order matters before you calculate anything. Then open the answer and compare both the method and the number. If you want a place to record the slips and come back to them later, the mistake log and retest queue is designed for that.
For help with this topic from a teacher who works with you one to one, see our online Additional Mathematics tuition.