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)