Combinatorics Decision Table
| Type | Order Matters? | Repetition Allowed? | Formula | Example | Key Insight |
|---|---|---|---|---|---|
| Permutations without Repetition | ✅ YES | ❌ NO | P(n,r) = n!/(n-r)! | How many ways can 3 people finish 1st, 2nd, 3rd in a race of 10 people? | Arranging distinct items in sequence |
| Permutations with Repetition | ✅ YES | ✅ YES | n^r | How many 4-digit PINs using digits 0-9? | Arranging items where same item can be reused |
| Combinations without Repetition | ❌ NO | ❌ NO | C(n,r) = n!/(r!(n-r)!) | How many ways to choose 3 people for a committee from 10 candidates? | Selecting distinct items (no arrangement) |
| Combinations with Repetition | ❌ NO | ✅ YES | C(n+r-1,r) | How many ways to buy 3 donuts from 5 different types? | Selecting items where same item can be chosen multiple times |
Quick Decision Process
Step 1: Ask yourself: “Does the order of selection/arrangement matter?”
- If someone swaps two items, does it create a different outcome?
- YES → Use Permutations
- NO → Use Combinations
Step 2: Ask yourself: “Can the same item be used/chosen multiple times?”
- YES → Use “with repetition” formula
- NO → Use “without repetition” formula
Memory Aids
- Permutations = Positions matter = Particular Placement
- Combinations = Collections = Choosing without Care for order
- With repetition = same item can appear multiple times
- Without repetition = each item used at most once
Common Keywords That Signal Each Type
| Permutations | Combinations |
|---|---|
| Arrange, Order, Sequence | Choose, Select, Committee |
| 1st/2nd/3rd place | Team, Group, Set |
| Passwords, Codes | Handshakes, Subsets |
| Line up, Rank | Menu items, Pizza toppings |