Order no · Repetition no
Combination without repetition
Use it to form groups where each element appears at most once. Example: choosing a committee.
Probability and Statistics
Combinations and permutations calculator
Answer two questions about order and repetition to choose the correct formula and obtain an exact integer result.
Choose the problem type
You do not need to memorize the formula. Indicate what changes a solution and the calculator will identify the case.
Formula and substitution
C(n, k) = n! / (k! × (n − k)!)
The substitution will appear after a valid calculation.
Procedimiento resumido
- Complete the inputs to see the procedure.
Publicidad
How to choose the correct formula
The calculator's two questions correspond to these four cases.
Order yes · Repetition no
Arrangement without repetition
Use it when positions are distinct and no element can repeat. Example: gold, silver and bronze.
Order no · Repetition yes
Combination with repetition
Use it to choose quantities where an option can appear several times. Example: scoops of ice cream by flavor.
Order yes · Repetition yes
Arrangement with repetition
Each position can use any option. Example: a PIN or a fixed-length password.
Examples to try
Each example activates a different combination of order and repetition.
Committee
Choose 3 people from 10 when order does not change the group.
n = 10, k = 3
Podio
Assign gold, silver and bronze among 10 participants without repetition.
n = 10, k = 3
Sabores
Choose 3 scoops from 5 flavors, allowing flavors to repeat.
n = 5, k = 3
PIN
Create a 4-position PIN using 10 digits that can repeat.
n = 10, k = 4
Large results without losing precision
Many combinatorial operations quickly exceed JavaScript's safe number range. This tool calculates with BigInt integers. Scientific notation only summarizes the display; the exact result remains available to view and copy.
Frequently asked questions
What is the difference between a combination and a permutation?
In a combination, order does not change the result. In a permutation or arrangement, changing the order creates a different result. Choosing Ana and Luis for a committee is the same selection as Luis and Ana; assigning them first and second place is not.
How do I know whether order matters?
Ask whether swapping two elements produces a different result. If it changes a podium, password or assignment of positions, order matters. If the final group stays the same, it does not.
What does allowing repeated elements mean?
It means the same option can be used more than once. Digits can repeat in a PIN; one person cannot occupy two places on a podium.
What happens if k is greater than n?
Without repetition, you cannot choose more elements than are available. With repetition, it may be valid because an option can be used several times.
Why do some results have so many digits?
The number of possibilities grows very quickly. The calculator preserves the exact integer with BigInt, shows a compact view and lets you copy every digit.
What is the difference between n^k and a permutation of repeated elements?
n^k counts sequences of k positions with n options at each position. Ordering a set that already contains repeated elements, such as the letters in MAMA, requires another formula and is outside this calculator's scope.