Permutations & Combinations
Explore Pascal’s triangle, visualize selections (C) and arrangements (P), and see how nPr and nCr are computed step by step.
Counting
Adjust n and r to explore counting
Values
| n | -- |
| r | -- |
| P(n,r) | -- |
| C(n,r) | -- |
| n! | -- |
| P/C ratio | -- |
The Math Behind It
Permutations
- Formula: P(n, r) = n! / (n − r)!
- Order matters — different arrangements of the same items count separately
- Multiplication principle: n × (n−1) × … × (n−r+1)
- Special case: P(n, n) = n! — total arrangements of all items
- Example: ways to award gold, silver, bronze from 10 athletes = P(10, 3) = 720
Combinations
- Formula: C(n, r) = n! / (r!(n − r)!)
- Order doesn’t matter — only the selection counts, not the arrangement
- Pascal’s Triangle: C(n, r) = C(n−1, r−1) + C(n−1, r)
- Binomial theorem: (a + b)n = ∑ C(n, k) an−k bk
- Relationship: P(n, r) = C(n, r) × r!
You're crushing it!