3.1 Fundamental Counting Principle
Key Takeaways
- The fundamental counting principle multiplies choices across connected stages of one outcome, but separate cases are added after each case is counted.
- Fill the most restricted slot first, especially when digits, people, or objects cannot repeat.
- A Math 30-2 counting answer often becomes the denominator or numerator of a later probability, so a small counting error can corrupt the whole solution.
- Use a complement when a restriction says at least one or not all; it is usually shorter than listing every successful case.
Why the counting principle matters
Probability is one of the largest Math 30-2 reporting areas, and many probability questions begin as counting questions. Before you can find a probability, odds ratio, or expected value, you often need to know how many total outcomes are possible and how many outcomes fit the condition. The fundamental counting principle says that if one task can be completed in a stages and the next task can be completed in b stages, then the combined task can be completed in a times b ways. With more stages, keep multiplying.
This rule is simple, but diploma questions rarely present it as a plain multiplication sentence. They describe licence plates, identification codes, schedules, menus, paths, teams, or passwords. Your job is to translate the wording into slots. A slot is one position or decision in the final outcome. If the outcome is a three-character code, there are three slots. If the outcome is an outfit made from a shirt, pants, and shoes, there are three slots.
Slot method
Use this routine before touching the calculator:
- State what a single outcome looks like.
- Draw or list one slot for each decision.
- Put restrictions above the affected slots.
- Fill the most restricted slot first.
- Multiply within one continuous case.
- Add only when you split into separate non-overlapping cases.
| Wording clue | Counting action | Watch for |
|---|---|---|
| and, followed by, then | Multiply the slots | Restrictions may change later slots |
| either case 1 or case 2 | Count each case, then add | Cases must not overlap |
| no repetition | Decrease choices after each use | Decide whether zero is allowed |
| at least one | Often use total minus none | Direct casework may be long |
| exactly one | Build a case with one success and the rest failures | Place the success in every possible position |
Worked example: restricted code
A student creates a four-digit access code using digits 0 through 9. The first digit cannot be 0, and no digit may repeat. How many codes are possible?
A single outcome is a four-slot code. The first slot has 9 choices because 1 through 9 are allowed. After the first digit is used, 9 digits remain for the second slot, including 0 if it was not already used. Then 8 digits remain, then 7. The count is 9 times 9 times 8 times 7 = 4536.
The common error is to write 10 times 9 times 8 times 7, which treats the first digit as if zero were allowed. Another error is 9 times 8 times 7 times 6, which wrongly removes zero from every later position. Restrictions apply only where the wording says they apply.
Worked example: separate cases
A cafeteria combo has one main, one side, and one drink. There are 5 mains, 4 sides, and 3 drinks. Two mains are vegetarian. How many combos are vegetarian or include iced tea, if iced tea is one of the drinks?
This is an or question with overlap. Count vegetarian combos: 2 times 4 times 3 = 24. Count iced tea combos: 5 times 4 times 1 = 20. Vegetarian iced tea combos were counted twice, so subtract 2 times 4 times 1 = 8. The total is 24 + 20 - 8 = 36.
That example shows why the fundamental counting principle and set logic connect. When cases overlap, adding raw counts double-counts shared outcomes.
Diploma traps
Trap 1: adding choices inside one outcome. If a locker code has 6 choices for the first dial and 5 for the second, the count is 30, not 11. Addition describes alternatives; multiplication describes combined decisions.
Trap 2: treating every stage as independent. In counting, a later stage may depend on an earlier one because an item has been removed. Do not confuse this with independent events in probability. Here, the practical question is whether the number of choices changes after a selection.
Trap 3: counting the complement but forgetting to subtract. For at least one repeated feature, count all outcomes, count outcomes with none of that feature, then subtract from the total. The complement is not the answer unless the question asks for none.
Trap 4: rounding too early. If a count becomes part of a probability, keep the exact whole-number count until the final probability is requested. Alberta numerical-response instructions commonly expect final rounding only when the question says so.
A strong scratch-work layout is a row of slots with numbers above them and a short note beside each restriction. That layout also helps written-response marking because another reader can see why each factor belongs in the product.
Numerical-response checkpoint
Counting questions may appear in a numerical-response format, so the final count has to fit the recording instruction. If the answer is a whole number, do not convert it to a decimal probability just because the topic is probability. If a later part asks for a probability, use the whole-number count as the numerator or denominator first, then simplify or round only as directed.
For ordered-response style items, keep the order exactly as requested. A count of 4536 and a code recorded as 4536 look similar, but one is a quantity and the other may be a sequence of digits. Labeling your scratch work as count, code, probability, or order prevents that mix-up.
A five-character school login uses one letter followed by four digits. The letter can be any of 26 letters. The first digit cannot be 0, and the four digits cannot repeat. How many logins are possible?
A survey prize code is made from three different letters chosen from A, B, C, D, and E. Which setup correctly counts the codes if order matters?
A meal has 4 mains, 3 sides, and 2 drinks. One drink is water. How many meals include at least one non-water drink?