Permutation & Combination
Calculate permutations and combinations with and without repetition
How to use Permutation & Combination
- 1Enter the total number of items n with the selected number k.
- 2Select the order (in the order of importance) or grouping (in the order not relevant).
- 3View the count and, if necessary, expand the formula.
Distinction between Arrange and Group
Arrange P(n,k) = n! / (n-k)!; Group C(n,k) = n! / (k! (n-k)!)Lists the order of order of calculation, such as the number of winners; groups that do not have the order of calculation, such as one hand.
Because the group ignores the order, the same n as k, C(n,k) is always less than or equal to P(n,k).
Arranges by multiplying the order of the rows to be "sequencedly removed k from n"; the number of combinations is divided by k! to remove the internal order of the k elements. So for the same n, k, grouping is always equal to ranking by k!
Repeat the selection equation: the number of n^k and the number of combinations is C (n+k-1, k) when repetition is allowed. The draw (non-release) is typically a non-repeat combination, while “one course per day from the five dishes, repeatable” is a repetitious combination.
| n (total) | k (selected) | Arrange P(n,k) | Group C(n,k) | Annotations |
|---|---|---|---|---|
| 5 | 2 | 20 | 10 | The order is the order. |
| 5 | 3 | 60 | 10 | The number of combinations is equal to k! |
| 10 | 2 | 90 | 45 | C = P ÷ k! |
| 8 | 4 | 1680 | 70 | k The greater the gap, the greater the gap. |
| 6 | 6 | 720 | 1 | There's only one combination for all. |
Different n, k to group
Frequently asked questions
When does it matter?
When the exchange of two of the options produces different results, such as ranking or password.
Why is it smaller?
Each group of k items corresponds to k!
What if k = n?
Both are equal to 1: There is only one way to select or sort all items.
How do we change the formula when we allow repetition?
Rearranged to n^k; regrouped to C(n+k−1, k), i. e. replace "k selected, allowed to repeat" with "k selected from n+k-15".
What's the "separation method"?
Repeatable combinations are often understood by using the partition method: divide k balls to the n class, with an equal value of k partitions at n+k-1 intervals.