Combinations & Permutations Calculator
How It Works
This combinations & permutations calculator uses established formulas to provide accurate results.
The basic rule:
- C(n,r) = n! / (r! × (n-r)!)
- P(n,r) = n! / (n-r)!
- n! = n × (n-1) × ... × 2 × 1
- 0! = 1
Results are estimates. Consult a professional for critical decisions.
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Explore Allday Everyday Math →Frequently Asked Questions
What is the difference between combinations and permutations?
Combinations count the number of ways to choose items where order does not matter (e.g., selecting a committee). Permutations count arrangements where order matters (e.g., ranking contestants). nPr is always ≥ nCr.
What is the formula for combinations?
C(n,r) = n! / (r! × (n-r)!). For example, C(10,3) = 10! / (3! × 7!) = 120. This counts how many ways to choose 3 items from 10 without regard to order.
What is the formula for permutations?
P(n,r) = n! / (n-r)!. For example, P(10,3) = 10! / 7! = 720. This counts how many ways to arrange 3 items chosen from 10 where order matters.
What is a factorial?
n! (n factorial) is the product of all positive integers from 1 to n. For example, 5! = 5×4×3×2×1 = 120. By convention, 0! = 1.
When do I use combinations vs permutations in probability?
Use combinations when the outcome set is unordered (lottery picks, card hands, committees). Use permutations when order matters (passwords, race placements, seating arrangements).