Permutations and combinations are two ways to count. Permutations count arrangements where order matters, and combinations count selections where order does not.
This chapter sits inside the SPM Additional Mathematics guide. Its ideas also feed into probability distributions, where nCr appears in the binomial formula.
How do the four skills fit together?
Each counting question uses the same thinking in the same order. The lessons follow it.
- Deciding whether order matters: choose between nPr and nCr.
- Counting arrangements with restrictions: blocks, fixed positions and forbidden cases.
- Counting selections with required members: people who must be in, out or split by type.
- Distinguishing over-counting from under-counting: checking that each outcome is counted exactly once.
A short orienting example
Take three students, Aini, Bala and Chen. As a line for a photo, there are 3! = 6 arrangements: ABC, ACB, BAC, BCA, CAB and CBA. As a group of three chosen from three, there is only 1 selection.
The same people, two different answers, because in a line the order counts and in a group it does not. Every question in this chapter starts by asking which of the two is happening.
Who should start where?
Pick your entry point by the mistake you make:
- If you tend to pick the wrong formula, begin with the order lesson.
- If your formula is right but your answers are too large, go to over-counting.
- If “at least” or “must include” questions confuse you, go to required members.
- If you want a timed check, try the permutations and combinations practice set.
The word-problem structure worksheet helps split a long question into parts. For a teacher who can question your reasoning directly, see online one-to-one Additional Mathematics tuition.