Combinatorics Lab

Parameters

n: 5

k: 2

n! = 120

C(5,2) = 10

P(5,2) = 20

Catalan(2) = 2

Combinatorics Explorer – Counting, Permutations & Combinations

Combinatorics is the branch of mathematics concerned with counting, arrangement, and combination of discrete objects—the foundation of probability theory, algorithm analysis, and cryptography. This explorer lets you compute permutations and combinations for sets you define, visualize Pascal's triangle and its diagonal properties, and trace the binomial theorem coefficient by coefficient. Seeing these structures visually makes abstract formulas like n-choose-k feel as natural as counting items in a list.

What you can do in this simulation

Compute nPr and nCr for user-defined n and r and see the formula breakdown
Build Pascal's triangle row by row and highlight binomial coefficients for any expansion
Enumerate all permutations of a small set and count them to verify the factorial formula
Explore the pigeonhole principle with configurable box and item counts
Visualize inclusion-exclusion with overlapping sets and count union sizes

Concepts covered

combinatorics · permutations · combinations · Pascal's triangle · binomial theorem · inclusion-exclusion

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