Combinatorics & Permutations
Master counting complexity. Spin combination cylinders, trace branching choice trees, and study the explosion of combinatorial permutations.
Vault dial
Safe Combination Lock
Permutations describe arrangements where order matters (e.g. PIN codes), while Combinations describe arrangements where order is irrelevant (e.g. lottery picks). Spin the safe dial wheels to see possible code permutations.
Combinatorial Space
Space: 8³ = 512 codes
Combinatorial Formula:
Formula: rⁿ = 8³
Calculation: 8 × 8 × 8 = 512
Each additional dial multiplies the overall choice space exponentially.
Calculation: 8 × 8 × 8 = 512
Each additional dial multiplies the overall choice space exponentially.
Graph Mapping
Choice Tree Branching
Trace how choices fork step-by-step. The tree below draws all possible coordinate decision paths for a mini-lock of n dials with r choices.
Discrete Permutations Matrix
Permutations with Repetition:
Order matters; items can be reused (e.g. padlock codes 7-7-7).
Formula: rⁿ
Formula: rⁿ
Permutations without Repetition:
Order matters; items cannot be reused (e.g. race finishes 1st, 2nd, 3rd).
Formula: P(r, n) = r! / (r - n)!
Formula: P(r, n) = r! / (r - n)!
Combinations (No Repetition):
Order is irrelevant; items cannot be reused (e.g. card hands).
Formula: C(r, n) = r! / [n! · (r - n)!]
Formula: C(r, n) = r! / [n! · (r - n)!]
Self Evaluation
Counting Quiz
Assess your understanding of factorials, combination counts, and permutation lock states.
Question 1 of 5
Score: 0/0
What is the main mathematical difference between a Permutation and a Combination?
Correct Answer!
Explanation goes here.
Focus Area Vocabulary
- Factorial (!): n! = n × (n-1) × ... × 1
- Permutation: Order MATTERS (e.g. Lock code)
- Combination: Order DOES NOT matter (e.g. Team picks)