Permutation Calculator
Calculate permutations ($P(n, r) = \frac{n!}{(n - r)!}$), permutations with repetition ($n^r$), circular permutations ($(n-1)!$), and combinations.
Calculate Ordered Permutations P(n, r)
Enter total available items (n) and number of chosen items (r).
Calculation Results
Calculated using permutation factorial equations: P(n, r) = \frac{n!}{(n - r)!}
Quick Summary
The Permutation Calculator evaluates ordered arrangements $P(n, r) = \frac{n!}{(n - r)!}$, permutations with repetition ($n^r$), circular permutations ($(n-1)!$), and combinations.
Formula Explanation
P(n, r) = \frac{n!}{(n - r)!}
\text{Permutations with Repetition} = n^r
\text{Circular Permutations} = (n - 1)!How It Works
A permutation is an ordered arrangement of set elements. The Permutation Calculator evaluates $P(n, r)$ using factorials, computing distinct ordered sequences when order matters.
Step-by-Step Worked Example
Practical Problem: Calculate the number of permutations when choosing $r = 3$ items from $n = 5$ available items.
- Step 1: Calculate numerator factorial ($n! = 5!$): $5! = 5 \times 4 \times 3 \times 2 \times 1 = \mathbf{120}$.
- Step 2: Calculate denominator factorial ($(n - r)! = 2!$): $2! = 2 \times 1 = \mathbf{2}$.
- Step 3: Divide numerator by denominator: $P(5, 3) = 120 / 2 = \mathbf{60\text{ permutations}}$.
- Step 4: Calculate Permutations with Repetition ($n^r$): $5^3 = 5 \times 5 \times 5 = \mathbf{125}$.
- Step 5: Interpretation: There are 60 distinct ordered arrangements of 3 items chosen from 5 items without repetition.
Real-World Calculation Examples
Scenario 1: 3 Items from 5 (P(5, 3))
Parameters: n = 5, r = 3
Result: P(5, 3) = 60 permutations (C(5, 3) = 10 combinations).
Scenario 2: Lock Combination Code (3 digits 0-9 with repetition)
Parameters: n = 10, r = 3 with repetition
Result: 10^3 = 1,000 codes.
Scenario 3: Race Gold, Silver, Bronze Winners (n = 8 runners)
Parameters: n = 8, r = 3
Result: P(8, 3) = 336 podium finishes.
Scenario 4: Seating 6 People at a Round Table
Parameters: n = 6 circular
Result: (6 - 1)! = 120 circular arrangements.
Key Benefits of Using This Calculator
Ordered Permutations $P(n, r)$
Computes exact ordered arrangements where item sequence matters.
Repetition & Circular Permutations
Outputs permutations with replacement ($n^r$) and circular table arrangements ($(n-1)!$).
Combination Comparison $C(n, r)$
Displays unordered combinations $C(n, r)$ alongside permutations.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a permutation?
A permutation is an arrangement of items where sequence/order is important ($P(n, r) = \frac{n!}{(n - r)!}$).
What is difference between permutation and combination?
Permutations care about ORDER (e.g. passcode 123 $\neq$ 321); combinations do NOT care about order (e.g. fruit salad of apples and bananas).
How do I calculate permutations in Excel?
Use formula =PERMUT(n, r) for no repetition or =PERMUTATIONA(n, r) for repetition allowed.
What is circular permutation formula?
Circular permutations equal $(n - 1)!$ because rotating seating positions does not create a new relative arrangement.
What are permutations with duplicate identical items?
Formula is $\frac{n!}{n_1! n_2! \dots n_k!}$ (e.g. arrangements of letters in "MISSISSIPPI").
What is $P(n, n)$ equal to?
$P(n, n) = n!$ (permuting all $n$ items).
What is $P(n, 0)$ equal to?
$P(n, 0) = 1$ (1 way to select zero items).
What is $0!$ equal to?
$0! = 1$ by mathematical definition.
Why is a digit padlock called a "combination lock"?
Mathematically, a passcode lock is a PERMUTATION lock, because 1-2-3 is different from 3-2-1!
How do permutations relate to matrix determinants?
Matrix determinants use the Leibniz formula, summing signed permutations of matrix entries.