Combinatorics Visualization

Interactive exploration of combination concepts in probability

Pascal's Triangle

Each number is the sum of the two numbers above it. Row n and column k represent the binomial coefficient C(n, k). Hover over a number to see the relationship!

Number of Rows: 20

Lattice Paths

A particle at (0,0) wants to reach (n, k). It can only move Right (R) or Up (U). How many unique paths are possible?

X Coordinate (Right):

Y Coordinate (Up):

🎲 Random Path

Total steps = 5 (Right + Up)

Choose 2 'Up' steps from 5 steps

C(5, 2) = 10

10 Paths

Example of one path:

R → R → R → U → U

Handshake Problem

If there are n people in a room and everyone shakes hands once with everyone else, how many total handshakes occur?

Number of People: 20

Each handshake is a selection of 2 people from n people.

C(n, 2) = n(n-1)/2 = 5×4/2 = 10

10 Handshakes

People Count

5

Total Handshakes

10

✨ Calculate ✕ Clear

🔤

Enter a word to see permutation calculation results

Case: Adjacent Identical Letters

0

Identical letters are treated as one package (block).

Standard Permutation

0

Formula: n! / (k₁! × k₂! × ...)