Understanding your result
Combinations count groups without regard to order. Permutations count ordered arrangements of the same selection size. The third result is r!, the number of orders for any one selected group. Large counts are displayed as complete integers rather than rounded scientific notation.
After identifying favorable and total equally likely outcomes, use the Probability Calculator to express their ratio as a chance.
The formula
n is the total number of distinct available items and r is the number selected. Both are whole numbers with 0 ≤ r ≤ n. Dividing permutations by r! removes the multiple orderings of each group. The empty product and 0! both equal one.
Selecting three people from ten
With n = 10 and r = 3, there are 120 three-person groups. Assigning three different positions creates 720 ordered arrangements. Each group has 3! = 6 orders, and 120 × 6 = 720.
How to use this calculator
- Count the distinct available items and enter n.
- Enter how many items are chosen as r.
- Use combinations for groups, or permutations when positions or order matter.
A group and a ranked list are different outcomes
Choosing three committee members ignores the order in which their names were drawn. Choosing a president, secretary, and treasurer assigns different roles, so exchanging two names creates another outcome. These tasks use combinations and permutations respectively.
The items must be distinguishable. Arranging repeated letters in a word needs a repeated-item adjustment and is not the same problem as choosing distinct people.
Without replacement is an essential condition
Once an item is selected, this model cannot select that same item again. A three-digit code that allows repeated digits does not fit this calculation; its number of possibilities comes from separate choices at each position.
Selecting zero items yields one empty selection. Selecting all n items yields one combination but n! different orders. These boundary results follow the same counting definitions as ordinary selections.
Assumptions & limitations
What this calculation assumes
- Items are distinct and each can be selected at most once.
- Counts are whole numbers between zero and 200.
What to keep in mind
- Repeated objects, replacement, circular arrangements, and other restrictions require different models.
- A count of possibilities alone does not establish that every outcome is equally likely.
Common questions
Which result should I use for a lottery selection?
Use combinations only if the selection ignores order and draws distinct items without replacement. Bonus pools and other rules may require additional calculations.
Why is n choose zero equal to one?
There is one way to select nothing: the empty group.
Are very large results rounded?
No. These counts use exact integer arithmetic and are shown in full within the supported input limit.