Combinations & Permutations Calculator

Calculate combinations (nCr), permutations (nPr), and factorials with step-by-step solutions. Perfect for probability, statistics, and counting problems.

Combinations & Permutations

C(n, r) = n! / (r!(n-r)!)

With Repetition (Replacement)

C(n+r-1, r) = (n+r-1)! / (r!(n-1)!)

Circular Permutations

(n-1)!

Arrangements around a circle where rotations are considered the same

Real-World Examples

Lottery Odds (6/49)
What are the odds of winning a lottery where you pick 6 numbers from 49?
Uses: Combinations C(49,6)
Password Possibilities
How many 8-character passwords using lowercase letters?
Uses: Permutations with Rep (26^8)
Seating Arrangements
How many ways can 8 people sit around a round table?
Uses: Circular Permutations (7!)
Poker Hand
How many possible 5-card poker hands from a 52-card deck?
Uses: Combinations C(52,5)

Formulas Reference

Combinations (nCr)
C(n,r) = n! / (r!(n-r)!)
Order doesn't matter. Selecting r items from n items.
Permutations (nPr)
P(n,r) = n! / (n-r)!
Order matters. Arranging r items from n items.
Combinations with Repetition
C(n+r-1, r)
Selecting r items from n types, items can repeat.
Permutations with Repetition
n^r
Arranging r items from n types, items can repeat.
Circular Permutations
(n-1)!
Arranging n items in a circle (rotations are same).
Factorial
n! = n x (n-1) x ... x 1
Product of all positive integers up to n. 0! = 1.

Frequently Asked Questions

What is the difference between combinations and permutations?

Combinations count selections where order does not matter (like choosing lottery numbers), while permutations count arrangements where order matters (like assigning race positions). For the same n and r, permutations always give a larger or equal result because each combination can be arranged in multiple ways.

How do you calculate C(n,r) combinations?

The combination formula is C(n,r) = n! / (r!(n-r)!), where n is the total items and r is items chosen. For example, C(5,2) = 5! / (2! * 3!) = 120 / (2 * 6) = 10. This counts the number of ways to choose r items from n items when order doesn't matter.

What are lottery odds and how are they calculated?

Lottery odds are calculated using combinations since the order of drawn numbers doesn't matter. For a 6/49 lottery, the odds are C(49,6) = 13,983,816 to 1. This means there are nearly 14 million possible combinations, making any single ticket extremely unlikely to win.