Counting Techniques
Master the fundamental principles of counting: multiplication rule, permutations, combinations, and arrangements with non-distinct items.
Multiplication Rule for Counting
If a task consists of a sequence of choices, the total number of ways to complete the task is the product of the number of choices at each step.
🔍 LogicLens: When to Use
Use for "tree-style decisions" where you make multiple independent choices in sequence: choosing an outfit, creating a meal combo, or building a password.
Example: Creating a Meal Combo
A restaurant offers 4 entrees, 3 sides, and 2 desserts. How many different meal combos can you create?
Solution:
Permutations (Order Matters)
A Permutation is an ordered arrangement of objects chosen from distinct objects without replacement.
(where )
⚠️ The Order Rule
"ABC" is a DIFFERENT permutation than "CBA". Order matters! Use permutations for race finishes, officer elections, or ranking competitions.
Example: Race Podium
In a race with 10 runners, how many ways can Gold, Silver, and Bronze medals be awarded?
Solution:
There are 720 different ways to award the top 3 medals.
Combinations (Order Does NOT Matter)
A Combination is a selection of objects from distinct objects without replacement, where order is irrelevant.
(also written as or "n choose r")
✅ The Group Rule
"ABC" and "CBA" are the SAME combination. Order doesn't matter! Use combinations for committee selection, card hands, or lottery picks.
🔀 Permutation vs Combination
Permutation
- ✓ Order MATTERS
- ✓ ABC ≠ CBA
- ✓ Use: Rankings, passwords
- ✓ Formula:
Combination
- ✓ Order does NOT matter
- ✓ ABC = CBA
- ✓ Use: Committees, lottery
- ✓ Formula:
Example: Pizza Toppings
A pizza shop offers 12 toppings. If you want to choose 3 different toppings, how many different pizzas can you create?
Solution:
You can create 220 different pizzas!
Permutations with Non-Distinct Items
When some items are identical, we must divide by the factorials of the number of identical items to avoid overcounting.
where are the counts of each identical item
Example: Arrangements of "STATISTICS"
How many different "words" can be formed using all the letters in STATISTICS?
Step 1: Count each letter
- • S appears 3 times
- • T appears 3 times
- • I appears 2 times
- • A appears 1 time
- • C appears 1 time
- • Total letters: 10
Step 2: Apply the formula
There are 50,400 different arrangements!
Interactive Counting Tool
Counting Calculator
Calculator Logic Guide
Use Combinations when order doesn't matter (e.g., selecting a committee).
The total number of distinct objects in the set.
How many items you are pulling out or arranging.
Adaptive Assessment
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