Comprehensive Calculator Guide
Permutations count the ways to pick r of n items and arrange them, when order matters and nothing repeats: P(n, r) = n! ÷ (n − r)!. The top three places among 10 runners can fall in 10 × 9 × 8 = 720 ways. Enter n and r for an exact answer, even for huge numbers.
How the count works
Fill the slots one at a time: 10 choices for first place, 9 left for second, 8 for third, so 720 in total. The formula n! ÷ (n − r)! is shorthand for that product. Arranging all n items gives n!: five books on a shelf, 5! = 120. By definition 0! = 1.
Ask whether swapping the order changes the outcome. In a race it does, so use permutations; for a committee it does not, so use combinations. That is why P(10, 3) = 720 while C(10, 3) = 120: each group of 3 can be ordered 3! = 6 ways.
With repetition
This calculator assumes no repetition. If items may repeat, as in a 4-digit PIN, the count is nʳ: 10⁴ = 10,000. For words with repeated letters, divide n! by the factorial of each repeat count: a 5-letter word with one letter twice has 5! ÷ 2! = 60 arrangements.
How to Use
- 01Enter n, the total number of items.
- 02Enter r, how many to pick and arrange (0 ≤ r ≤ n).
- 03Read the exact count; use the combination calculator if order does not matter.
Formula Used
P(n, r) = n! ÷ (n − r)! = n × (n − 1) × … × (n − r + 1)Practical Examples
Example 1: race places
10 runners, 3 places: 720 orders.
Example 2: code with distinct digits
4 different digits from 0–9: 10 × 9 × 8 × 7 = 5,040 codes.
Example 3: books on a shelf
5 different books: 5! = 120 arrangements.
Important Tips
- For small numbers, multiply r descending numbers starting at n.
- Results are exact up to n = 1,000.
Common Mistakes to Avoid
- Using permutations when order does not matter.
- Computing n! ÷ r! instead of n! ÷ (n − r)!.
Frequently Asked Questions
- What is the difference between permutations and combinations?
- Permutations count ordered arrangements; combinations count groups where order does not matter.
- How do I calculate permutations?
- Multiply r descending numbers from n: P(8, 3) = 8 × 7 × 6 = 336.
- What about repetition?
- If items can repeat, the count is nʳ.
- What is 0!?
- 1, by definition, which keeps P(n, n) = n!.