📖 The Big Idea
Counting problems on the GED ask: how many different combinations or arrangements are possible? These problems look complicated but they follow a simple pattern: set up the problem, then multiply.
There are two main types you'll see:
Type 1 — Counting Principle
One choice × another choice
Picking one item from each of several categories. Choices stay the same size. Example: 3 shirts × 4 pants × 2 shoes.
Type 2 — Arrangements (Permutations)
Choices shrink as slots fill up
Filling positions where order matters and each person/item can only be used once. Example: 10 runners racing for 1st, 2nd, 3rd.
The golden rule: When you make one choice AND THEN another choice, you almost always MULTIPLY — not add.
✖️ The Counting Principle
If one choice has a options and the next choice has b options, the total combinations = a × b.
This works for as many categories as you need. Just multiply all of them together.
The Formula
Total combinations = Choice 1 × Choice 2 × Choice 3 × ...
The hard part is recognizing which categories to multiply. The calculator does the rest.
Example: A restaurant offers 5 drinks, 6 entrees, and 4 desserts. How many meal combinations?
Calculator reminder: Type 5 × 6 × 4 into your calculator. Don't try to do this in your head.
🔢 Arrangements — When Choices Shrink
In some problems, order matters and each person (or item) can only be used once. As each slot gets filled, the number of remaining choices shrinks by one.
Key signal: If you see positions like 1st/2nd/3rd, President/VP/Treasurer, or any situation where "who fills which role" matters — choices shrink.
Example: A race has 10 runners. How many ways can 1st, 2nd, and 3rd place be awarded?
One slot at a time. Once a runner wins 1st place, they're gone. 10 choices shrink to 9, then to 8. Use your calculator: 10 × 9 = 90, then 90 × 8 = 720.
⚠️ Common Mistakes
Mistake 1
Adding instead of multiplying
5 shirts + 4 pants = 9? No — it's 5 × 4 = 20. One choice AND THEN another = multiply.
Mistake 2
Forgetting choices shrink
Writing 10 × 10 × 10 when a person can only fill one role. Choices shrink: 10 × 9 × 8.
Mistake 3
Disorganized setup
Jumping to multiply before listing the slots. Always identify each slot first, then fill in the numbers.
Mistake 4
Multiplying categories not chosen
If a customer skips entrees, don't multiply entrees. Only multiply the categories actually being chosen.
🔢 Worked Examples
Example 1 — Counting Principle
A store sells shirts in 6 colors, pants in 4 styles, and hats in 3 colors. How many different outfits are possible?
Step 1: List the categories: Shirts (6), Pants (4), Hats (3)
Step 2: One choice AND THEN another → multiply all three
Step 3: Use calculator: 6 × 4 = 24, then 24 × 3 = 72
✓ Answer: 72 outfit combinations
Example 2 — Arrangements (Shrinking Choices)
A club has 8 members. How many ways can a President and Vice President be selected? (No one can hold both roles.)
Step 1: 2 slots to fill — President, then VP
Step 2: President: 8 choices. After that, one person is used → VP: 7 choices
Step 3: Use calculator: 8 × 7 = 56
✓ Answer: 56 arrangements
Example 3 — Password/Code
A 4-letter code uses letters from A, B, C, D, E, F (6 letters). No letter can repeat. How many codes are possible?
Step 1: 4 slots. Letters can't repeat → choices shrink
Step 2: Slot 1: 6, Slot 2: 5, Slot 3: 4, Slot 4: 3
Step 3: Calculator: 6 × 5 = 30, × 4 = 120, × 3 = 360
✓ Answer: 360 possible codes
Example 4 — Multiple Categories
A vacation package offers 4 flights, 3 hotels, 5 activities, and 2 meal plans. How many total packages are possible?
Step 1: 4 categories, one choice from each → multiply all four
Step 2: 4 × 3 × 5 × 2
Step 3: Calculator: 4 × 3 = 12, × 5 = 60, × 2 = 120
✓ Answer: 120 vacation packages
✏️ Practice Questions