Understanding Combinations & Permutations
The Key Difference
The fundamental distinction is whether order matters. In combinations, selecting items A, B, C is the same as selecting C, B, A. In permutations, ABC is a different arrangement from CBA.
Combinations
Order does NOT matter. Used when selecting a group.
Example: Choosing 3 students from 10 for a committee
Permutations
Order MATTERS. Used when arranging in sequence.
Example: Assigning 1st, 2nd, 3rd place from 10 contestants
With vs Without Repetition
Without Repetition
Each item can only be selected once. Once picked, it's gone.
Example: Dealing cards from a deck
With Repetition
Items can be selected multiple times. Think of categories or types.
Example: Digits in a PIN code (can repeat)
The Four Counting Modes
Combinations without Repetition (nCr)
Formula: C(n, r) = n! / (r!(n-r)!)
Use: Selecting r items from n distinct items, order doesn't matter.
Example: Picking 5 cards from a 52-card deck for a hand.
Permutations without Repetition (nPr)
Formula: P(n, r) = n! / (n-r)!
Use: Arranging r items from n distinct items in order.
Example: Arranging 3 books on a shelf from 10 books.
Combinations with Repetition
Formula: C(n+r-1, r) (stars and bars)
Use: Selecting r items from n types when repeats are allowed.
Example: Choosing 5 donuts from 4 flavors.
Permutations with Repetition
Formula: nr
Use: Each of r positions can be any of n items.
Example: Creating a 4-digit PIN using digits 0-9.
Decision Tree
Does the order of selection matter?
Yes
→ Permutation
No
→ Combination
Can items be repeated?
Yes
→ With Repetition
No
→ Without Repetition
Important Notes
Factorial growth: These numbers grow extremely fast. 20! ≈ 2.4 × 1018. Large inputs may produce astronomically large results.
Relationship: P(n, r) = C(n, r) × r! — permutations count all orderings of combinations.
Special cases: C(n, 0) = 1 (one way to choose nothing), P(n, n) = n! (full arrangement), 0! = 1 by convention.
Example Calculation
Mode: Combinations without repetition (order doesn't matter)
Formula: C(49, 6) = 49! / (6! × 43!)
Calculation: = (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1)
Result: = 13,983,816
There are nearly 14 million different ways to choose 6 numbers from 49!
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