Permutations & Combinations
Solve permutations (nPr) and combinations (nCr) counting arrangements. Inspect factorial reductions and map results to Pascal's Triangle.
Total Arrangement Combinations
120
Alternate Counting Perspectives
Pascal's Triangle Alignment Map (Row n = 10)
Pascal row highlighted bubble matches combinations select parameter index (r = 3).
Factorial Cancellation Steps
Understanding Permutations vs. Combinations
In combinatorics, the central question is determining how many ways objects can be grouped or arranged. The most critical factor is order:
- Permutations (nPr): The order of selection matters. Arranging three letters A, B, and C gives 6 distinct outcomes: ABC, ACB, BAC, BCA, CAB, CBA.
- Combinations (nCr): The order of selection does not matter. Grouping three letters A, B, and C yields only 1 distinct group: '{A, B, C}'.
Repetition and Special Forms
- With Repetition: When selected items are placed back in the pool for subsequent choices.
- Circular Permutations: Arranging items along a circular track, where cyclic shifts are identical (yielding (n-1)!).
- Multi-set Permutations: Arranging a set of objects containing indistinguishable duplicates (such as calculating word anagram possibilities).
Frequently Asked Questions About Combinatorics
What is the key difference between permutations and combinations?
The fundamental difference is order. In permutations, the order of selection matters (e.g., arrangements ABC and BCA are counted as different). In combinations, the order does not matter (e.g., groupings ABC and BCA are considered identical).
How does repetition change permutation and combination formulas?
When repetition is allowed, formulas change because items can be reused. Permutations with repetition becomes n^r. Combinations with repetition uses the stars and bars method, represented by the formula C(n+r-1, r).
What is a circular permutation?
Circular permutations calculate the number of ways to arrange n distinct objects in a circle. Because rotation is invariant, we fix one object and arrange the remaining n-1 objects, yielding the formula (n-1)!.
Can this handle circular permutations?
Yes. Calculate permutations (nPr), combinations (nCr), with or without repetitions.