Explore various combinatorics scenarios and their applications in problem-solving.
Arrangement of n distinct items in n places
Formula: n!
Example: For n=4: 4! = 24 arrangements
Use Case: Arranging books on a shelf, determining race finish orders
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Arrangement of n items where some are identical
Formula: n! / (n1! * n2! * ... * nx!)
Example: For n=4 with 2 A's and 2 B's: 4! / (2! * 2!) = 6 arrangements
Use Case: Arranging letters in words with repeated letters
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Picking r different items from n items where order matters
Formula: P(n,r) = n! / (n-r)!
Example: For n=5, r=3: 5! / (5-3)! = 60 arrangements
Use Case: Selecting and arranging podium finishers from a group of runners
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Picking r different items from n items where order doesn't matter
Formula: C(n,r) = n! / (r! * (n-r)!)
Example: For n=5, r=3: 5! / (3! * 2!) = 10 combinations
Use Case: Selecting a committee from a group of people
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Dividing n distinct items into r distinguishable groups
Formula: S(n,r) (Stirling number of the second kind)
Example: For n=4, r=2: S(4,2) = 7 partitions
Use Case: Dividing students into different classes
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Distribution of n different items into r numbered cells
Formula: r^n
Example: For n=3, r=2: 2^3 = 8 distributions
Use Case: Assigning tasks to specific days of the week
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Distribution of n identical items into r cells, empty cells allowed
Formula: C(n+r-1, r-1)
Example: For n=5, r=3: C(7,2) = 21 compositions
Use Case: Distributing identical candies among children, allowing some to receive none
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Distribution of n identical items into r cells, no empty cells
Formula: C(n-1, r-1)
Example: For n=5, r=3: C(4,2) = 6 compositions
Use Case: Distributing identical tasks among team members, ensuring everyone gets at least one
For more details, see the Combinatorics Table and the Combinatorics Diagram.
Arranging n different items in a circle
Formula: (n-1)!
Example: For n=4: 3! = 6 arrangements
Use Case: Seating arrangements around a circular table
For more details, see the Combinatorics Table and the Combinatorics Diagram.
This diagram provides a visual overview of the relationships between different combinatorics scenarios. For detailed formulas and examples, refer to the Combinatorics Table below or the individual sections above.
| Scenario | Description | Formula | Example | Use Case |
|---|---|---|---|---|
| Permutation (Full) | Arrangement of n distinct items in n places | n! | For n=4: 4! = 24 arrangements | Arranging books on a shelf, determining race finish orders |
| Permutation with Repetition | Arrangement of n items where some are identical | n! / (n1! * n2! * ... * nx!) | For n=4 with 2 A's and 2 B's: 4! / (2! * 2!) = 6 arrangements | Arranging letters in words with repeated letters |
| Permutation without Repetition | Picking r different items from n items where order matters | P(n,r) = n! / (n-r)! | For n=5, r=3: 5! / (5-3)! = 60 arrangements | Selecting and arranging podium finishers from a group of runners |
| Combination | Picking r different items from n items where order doesn't matter | C(n,r) = n! / (r! * (n-r)!) | For n=5, r=3: 5! / (3! * 2!) = 10 combinations | Selecting a committee from a group of people |
| Partition into Groups | Dividing n distinct items into r distinguishable groups | S(n,r) (Stirling number of the second kind) | For n=4, r=2: S(4,2) = 7 partitions | Dividing students into different classes |
| Distribution into Cells | Distribution of n different items into r numbered cells | r^n | For n=3, r=2: 2^3 = 8 distributions | Assigning tasks to specific days of the week |
| Weak Composition | Distribution of n identical items into r cells, empty cells allowed | C(n+r-1, r-1) | For n=5, r=3: C(7,2) = 21 compositions | Distributing identical candies among children, allowing some to receive none |
| Strong Composition | Distribution of n identical items into r cells, no empty cells | C(n-1, r-1) | For n=5, r=3: C(4,2) = 6 compositions | Distributing identical tasks among team members, ensuring everyone gets at least one |
| Circular Permutation | Arranging n different items in a circle | (n-1)! | For n=4: 3! = 6 arrangements | Seating arrangements around a circular table |
For more detailed explanations of each scenario, click on the scenario names in the table or refer to the individual sections above. You can also check the Combinatorics Diagram for a visual representation.