Solve permutation, combination, and constraint satisfaction problems. Calculate possible arrangements, team selections, scheduling scenarios, and probability-based outcomes. This calculator helps solve real-world discrete math problems.
Select Problem Type
Permutation Calculator: P(n, r)
Permutations: Calculate ordered arrangements where ORDER matters. Used for ranking, seating, or sequential selection.
Combination Calculator: C(n, r)
Combinations: Calculate unordered selections where ORDER doesn't matter. Used for team selection, lottery, or subset choosing.
Team Selection with Constraints
Select team members from a pool with role requirements: Forward, Midfielder, Defender, Goalkeeper positions.
Scheduling & Assignment Problem
Assign workers to shifts: Calculate possible shift assignments given constraints (max shifts per person, required coverage).
Solution & Analysis
Understanding Combinatorics
What is combinatorics? Combinatorics is the mathematical study of counting, arranging, and selecting objects under various constraints. It answers questions like "In how many ways can we arrange these items?" or "How many possible combinations are there?"
Key Concepts
1. Permutations: Order Matters
A permutation is an ordered arrangement of items. The number of permutations of n items taken r at a time is:
Example: Podium positions (1st, 2nd, 3rd place) in a race with 10 runners:
A factorial (n!) is the product of all positive integers up to n:
Example: 5! = 5 × 4 × 3 × 2 × 1 = 120
Real-World Applications
Team Selection Problem
A sports manager has 8 forwards, 10 midfielders, 9 defenders, and 3 goalkeepers. The manager must select a team of 11: 3 forwards, 4 midfielders, 3 defenders, and 1 goalkeeper. How many different teams can be formed?
Calculation:
C(8, 3) = 56 ways to select 3 forwards from 8
C(10, 4) = 210 ways to select 4 midfielders from 10
C(9, 3) = 84 ways to select 3 defenders from 9
C(3, 1) = 3 ways to select 1 goalkeeper from 3
Total Teams = 56 × 210 × 84 × 3 = 3,111,696 possible teams
Scheduling Problem
Assign 5 workers to 3 shifts (morning, afternoon, evening), with 2 workers required per shift and maximum 2 shifts per worker. This is a constrained assignment problem solved using combinatorial methods.
Permutations vs Combinations Quick Reference
Aspect
Permutation
Combination
Order Importance
Order matters (ABC ≠ BAC)
Order doesn't matter (ABC = BAC)
Formula
P(n, r) = n! / (n-r)!
C(n, r) = n! / (r!(n-r)!)
Use Cases
Rankings, seating, passwords, sequences
Selections, teams, committees, subsets
Example: n=10, r=3
720 (much larger)
120 (smaller)
Worked Example: Lottery Problem
Scenario: A lottery requires selecting 6 numbers from 1-49. What are the odds of winning?
Solution:
This is a combination problem (order doesn't matter)
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