Counting cases means figuring out how many different ways something can happen.
Tossing a coin gives 2 cases, heads or tails, and rolling a die gives 6, the numbers 1 through 6.
When you decide things in sequence, like picking a top and then a bottom, you multiply the cases to count them all without missing any.
Here you can change the number of coins, the die, and the outfit combinations, and count the cases with a tree diagram and connecting lines.
What is "Counting Cases"?
🎲 All Possibilities That Can Happen!
①Coin toss → heads or tails = 2 cases
②Rolling a die → 1–6 = 6 cases
③'How many possibilities?' = counting cases!
2
seegongsik.com
Visualize the number of cases for coins and dice
Multiplication Rule
👕 Counting Outfit Combinations
①3 tops × 3 bottoms
②First top with 3 bottoms → 3
③Each top has 3 → 3 × 3 = 9 total!
3
seegongsik.com
Show all top/bottom outfit combinations as connecting lines
Summary
Sum Rule
A or B = (cases of A) + (cases of B)
Add when 'or' (mutually exclusive)
Product Rule
A and B = (cases of A) × (cases of B)
Multiply when 'and' (independent/sequential)
Cases for n Coins
2n (multiply 2 by itself n times)
1 coin = 2, 2 = 4, 3 = 8, 4 = 16…
🎯 Key Points
①Counting cases = enumerating all possibilities
②'or' → add (sum rule)
③'and'/'in sequence' → multiply (product rule)
④Tree diagram covers all cases
⑤n coins → 2ⁿ; 1 die → 6
Examples & Unit Review
Example 1
You have 3 tops and 4 bottoms. How many ways are there to get dressed?
1
You choose a top and then a bottom, so use the product rule: 3 × 4.
2
3 × 4 = 12 ways.
▸ 12 ways
When you choose two things in sequence, multiply the numbers of cases.
Example 2
When you toss 3 coins at once, how many possible outcomes are there?
1
One coin has 2 outcomes (heads or tails). Three coins give 2 × 2 × 2.
2
2 × 2 × 2 = 8 outcomes.
▸ 8 outcomes
The number of outcomes for n coins is 2 multiplied n times, or 2ⁿ.
Unit Review
When you roll one die, how many cases give an even number?
①3
②2
③6
④4
▸ ① 3
1
A die shows 1, 2, 3, 4, 5, 6. The even numbers are 2, 4, 6.
2
There are 3 even numbers, so the number of cases is 3.