H2 Maths Counting: Decide What One Outcome Looks Like

Define one outcome before choosing a formula

In a public H2 Maths counting discussion, a student could follow worked solutions but found it difficult to form a solution when reading a new question. That is a useful distinction: recognising someone else's expression is different from deciding what an expression should count.

Begin by writing one sentence describing a single outcome. Is it a team of three people, three people occupying different roles, or five people standing in a line? The 2026 H2 Mathematics 9758 syllabus includes counting principles, permutations, combinations and arrangements with restrictions. The original examples below isolate the choices that connect those ideas to a problem.

Compare a team with three named roles

Suppose eight distinct students are available. To choose a team of three with no named positions, an outcome is simply a set of three students. Selecting A, B and C gives the same team as selecting C, A and B. The number of teams is 8C3 = 56.

Now appoint a chair, secretary and treasurer, with one person per role. An outcome must identify who holds each job. There are eight choices for chair, seven remaining choices for secretary, then six for treasurer. Multiplying gives 336 appointments. Swapping the chair and secretary changes the outcome even though the same three students are involved.

The ratio between these answers has a meaning. Each team of three has 3! = 6 assignments to the three roles, and 56 × 6 = 336. Dividing by 3! in the team problem removes six repeated orderings of every team. It is not a general instruction to divide whenever the wording includes “choose”.

Translate restrictions into a construction

Return to the team of three. If one particular student must be included, that student's place is already fixed. Choose the other two people from the remaining seven: 7C2 = 21. Writing 8C3 and then multiplying by something for the required person would count the selection process twice.

Next suppose two particular students cannot both be on the team. Count all 56 teams and subtract the teams containing both. Once those two are included, choose one additional member from the other six, giving six forbidden teams. The required total is 56 − 6 = 50. This permits teams containing neither of the two students and teams containing exactly one. “Cannot both” does not mean “exactly one”.

If the condition instead says exactly one must be included, choose which of the two is present, then choose two teammates from the other six. That gives 2 × 6C2 = 30. Stating the condition in ordinary words before calculating prevents these two different questions from being merged.

Explain what a block represents

For a separate example, five distinct students stand in a line and two named students must be adjacent. Treat those two temporarily as one block. There are then four units to arrange: the block and three individuals. These have 4! arrangements. The two students inside the block can swap places, giving 2 × 4! = 48 lines.

The factor of two matters because the internal order changes the line. The block method also depends on the requirement actually being adjacency. If the condition were “not adjacent”, subtract the 48 adjacent arrangements from all 5! = 120 arrangements to obtain 72. This is a complement argument with clearly identified allowed and forbidden outcomes.

Choose between adding and multiplying

Multiply when an outcome is built through successive choices, with each factor counting the available options at that stage. Add when the valid outcomes are split into separate cases that do not overlap. If the cases overlap, one outcome may enter the total more than once. Explain the split rather than assuming separate lines of working automatically represent separate cases.

For practice, take a question from the JC paper collection and write an outcome sentence before touching the calculator. Under every factor, name what is selected or arranged. Afterwards, shrink the problem to four or five objects and list outcomes to test the method. The goal is a count in which every valid outcome appears once.