💡 What You Will Learn
The Fundamental Counting Principle is one of the most useful rules in GED math. Whenever a task has multiple independent steps — choosing a letter AND a digit, picking an outfit AND shoes — you simply multiply the number of choices at each step to get the total number of possible outcomes.
If you need to make a series of independent choices, the total number of possible outcomes equals the product (multiplication) of the number of choices at each step.
Independent means one choice does NOT affect the other — picking a letter doesn't change how many digits are available.
Example: Naming an apartment needs TWO steps — choose a letter, then choose a digit.
Letters: A, B, C, D → 4 choices
Digits: 1–9 → 9 choices
4 × 9 = 36 total apartments
Steps are independent — choosing one thing doesn't reduce choices for the next thing.
Addition — which counts mutually exclusive groups, not combinations of choices.
For each of the 4 letter choices, there are 9 digit choices. So the total is:
(A1, A2, A3...A9) + (B1, B2...B9) + (C1...C9) + (D1...D9) = 9+9+9+9 = 36
Adding 9 four times = 4 × 9 = 36. Multiplication is just repeated addition!
| Situation | Step 1 | Step 2 | Total |
|---|---|---|---|
| Apartments | 4 letters | 9 digits | 4×9 = 36 |
| Coin + Die | 2 sides | 6 sides | 2×6 = 12 |
| Outfits | 5 shirts | 3 pants | 5×3 = 15 |
| Menu | 4 mains | 3 sides | 4×3 = 12 |
Total = 4 × 9 = 36
13 is a trap answer — it's 4 + 9 = 13. Adding gives the total number of individual letters and digits, not the number of combinations. Always multiply when combining choices from separate categories!
40 might come from confusing "digits 1 through 9" (which is 9 digits) with "digits 1 through 10" (10 digits). Count carefully: 1,2,3,4,5,6,7,8,9 = 9 digits, not 10!
A tree diagram maps out every possible choice at each step as branches. Count the leaves (endpoints) to get the total number of outcomes — it always matches the multiplication!
The 6 end points (A1, A2, A3, B1, B2, B3) are the leaves of the tree. Count them = 6 = 2 × 3. Tree diagrams always confirm the multiplication!
Tree diagrams are helpful for small problems (2–3 choices per step). For larger problems like Greg's apartments (4 × 9 = 36), just multiply — drawing 36 branches would take too long!
The most common GED trap is adding the choices. 4 letters + 9 digits = 13 is wrong. 4 × 9 = 36 is correct!
"1 through 9" = 9 digits. "0 through 9" = 10 digits. "1 through 10" = 10 digits. List them if unsure!
Draw boxes or slots for each decision. Label each box with how many choices it has. Then multiply all the boxes.
5 shirts × 3 pants × 2 shoes × 4 colors = 5 × 3 × 2 × 4 = 120. The rule works for any number of steps!
If letters/digits CAN repeat: each slot has the full number of choices. If they CANNOT repeat: each slot has one fewer choice than the last.
| Answer | How You'd Get It | Correct? |
|---|---|---|
| 36 | 4 × 9 = 36 ✅ Multiply the choices | ✅ CORRECT |
| 13 | 4 + 9 = 13 ❌ Adding instead of multiplying | ❌ Wrong |
| 40 | 4 × 10 = 40 ❌ Counting 10 digits instead of 9 | ❌ Wrong |
| 16 | 4 + 4 + 4 + 4 = 16? No clear logic ❌ | ❌ Wrong |