Counting problems often turn on one question: does changing the order create a different outcome? Choosing a team and assigning people to distinct roles are not the same task.
Count ordered roles
There are five people available for captain and deputy. Choose a captain in five ways, then a different deputy in four ways. There are 20 assignments.
Alex as captain and Bea as deputy is different from Bea as captain and Alex as deputy. The roles make order meaningful.
Count an unordered pair
Now choose any two of the same five people for a team with no roles. The ordered count of 20 counts each pair twice: Alex–Bea and Bea–Alex represent the same team.
Divide by two to get 10 teams. The division corrects a specific duplication; it is not an arbitrary rule to apply whenever a question mentions two people.
Check whether repetition is allowed
A three-position code uses digits 1 through 4, with repetition allowed. Each position has four choices, producing 4 × 4 × 4 = 64 codes.
If repetition is prohibited, the choices become four, then three, then two, giving 24 codes. A small wording change alters the counting structure.
Add alternatives, multiply stages
If two non-overlapping routes to an outcome are alternatives, their counts add. If an outcome requires one choice followed by another, counts often multiply. Be careful: adding overlapping cases can count the same outcome twice.
Verify with a smaller version
If a large problem feels abstract, reduce it to three people or two positions and list all outcomes. The smaller example can reveal whether order or repetition matters and whether a proposed formula double-counts.
Then return to the full problem and state the choices at each stage. A clear counting argument is more dependable than recognizing a formula shape without checking whether its assumptions match the question.