Combinatorics Online Calculator

Professional online calculators for permutations, combinations, variations and counting principles

Basic Counting Principles

Rule of Product (Multiplication Principle)
Fundamental counting principle - total count by multiplying individual options

Permutations (Arrangements)

Permutations without Repetition
Number of arrangements of n different objects - Factorial n!

Combinations (Selections)

Combinations without Repetition
Binomial coefficient C(n,k) - Number of k-selections from n without regard to order
Combinations with Repetition
Multiset coefficient - Stars and Bars method for combinations with repetition

Variations (Ordered Selections)

Variations without Repetition
k-Permutations V(n,k) - Ordered selection of k from n different objects
Variations with Repetition
Exponential principle n^k - Each position can be chosen independently

Special Applications

Activity Selection Problem
Optimization problem for selecting non-overlapping activities

About Combinatorics Calculators

Combinatorics is a fundamental branch of mathematics dealing with counting, arrangement, and selection of objects. These calculators help in understanding and computing:

  • Permutations - Arrangements of objects
  • Combinations - Selections without order
  • Variations - Ordered partial selections
  • Probability Theory - Counting favorable cases
  • Optimization Problems - Selection strategies
  • Cryptography - Key space calculation
Important Formulas
Permutations
P(n) = n!
All arrangements of n objects
Combinations
C(n,k) = n!/(k!(n-k)!)
k from n selected, order irrelevant
Variations
V(n,k) = n!/(n-k)!
k from n selected, order matters
With Repetition
Variations: n^k
Combinations: C(n+k-1,k)
Tip: Combinatorics is fundamental for probability theory, statistics, computer science and many optimization problems in science and technology.
Quick Reference
4! = 24
Permutations
C(8,3) = 56
Combinations
V(8,3) = 336
Variations
3² = 9
With Rep.
a₁ × a₂ × ... × aₙ
Product Rule
Which Formula?
Order matters?
Yes: Permutations/Variations
No: Combinations
Use all objects?
Yes: Permutations (n!)
No: Variations/Combinations
Repetition allowed?
Yes: n^k or ((n,k))
No: V(n,k) or C(n,k)
Examples
Lottery 6 from 49:
C(49,6) = 13,983,816
Podium (1st, 2nd, 3rd):
V(10,3) = 720
4-digit PIN:
10⁴ = 10,000
Seating 6 people:
6! = 720
Related Calculator Categories
Mathematical Foundations
Applied Areas
All Areas