Example 1
- n = 5
- k = 3
60
What is V(5,3)? 60. Order on the podium counts.
nPr calculator
Type n and k. A variation V(n,k) counts ordered lists of k items from n. V(5,3) = 60. V(6,2) = 30. English textbooks usually say k-permutations or P(n,k).
Variations without repetition. When k is larger than n, the result is 0. V(n,n) = n!.
Results
Enter data and click Calculate.
How it works
A variation without repetition V(n,k) is the number of ways to line up k distinct items from a set of n. Order makes a new result. From five runners on a three-step podium V(5,3) = 60, because Anna-Bartek-Celina is not the same as Celina-Bartek-Anna.
The formula is n! / (n-k)!. You start from the full factorial n! and drop the orders of the people you did not pick. V(5,3) = 120 / 2 = 60. V(6,2) = 720 / 24 = 30. When k = n, the denominator is 0! = 1 and V(n,n) = n!.
The fields are n and k, whole numbers, n ≥ 0, k ≥ 0. The calculator uses the integer part. When k exceeds n, the result is 0, not a blank: you cannot seat seven distinct people from a set of five.
A combination C(n,k) also divides by k!, because it ignores order. V(5,2) = 20, C(5,2) = 10. Same pair, two line-ups. In English this formula is usually P(n,k) or a k-permutation. Variation is the school-Polish name kept on the page.
The header units switch does not shorten anything here. 5 and 3 stay 60 whatever the label says. This is a count of queues, not a length.
V(4,4) = 24. V(5,1) = 5. V(7,2) = 42. V(8,0) = 1, because of 0! in the denominator. V(3,5) = 0, because k > n.
Formula
V(n,k) = n! / (n-k)!, for 0 ≤ k ≤ n; when k > n the result is 0
How to use
- Type n, for example 5, and k, for example 3.
- Click Calculate. V(5,3) = 60.
- V(6,2) = 30. V(4,4) = 24, the same as 4!.
- When k is larger than n, the result is 0.
- Without order you count on combinations. Here A-B and B-A are two results.
V(5,3) = 60
A variation V(n,k) = n! / (n-k)!. V(5,3) = 60. V(6,2) = 30. Order counts. When k > n the result is 0.
- V(n,k)
- The number of ordered lists of k from n. V(5,3) = 60. V(n,n) = n!. Also written P(n,k).
- n
- The size of the set. With k = 3 and n = 5 you get 60. n smaller than k gives 0.
- k
- The length of the list. k = n returns the factorial. k = 0 gives 1.
Examples
Example 2
- n = 6
- k = 2
30
How many ordered pairs from 6? V(6,2) = 30.
Example 3
- n = 4
- k = 4
24
Is V(4,4) the same as 4!? Yes, 24.
Related calculators
Common questions
What is V(5,3)?
60. The formula 5! / 2! = 120 / 2 = 60. Three podium places from five people, and order counts.
What does variation without repetition mean?
You line up k distinct items from n and do not reuse a person. Order makes a new result. That is V(n,k) in this calculator.
How is V(n,k) different from C(n,k)?
A variation keeps order, a combination drops it. V(5,2) = 20, C(5,2) = 10. Each set has k! orders.
Is V(n,n) a factorial?
Yes. V(4,4) = 24 = 4!. When you take everyone, you are left with the permutations of n items.
What if k is larger than n?
The result is 0. You cannot line up more distinct items than you have in the set.
What do English books call V(n,k)?
Usually k-permutations or P(n,k). Same formula n! / (n-k)!. Variation is the Polish school name.
What are V(6,2) and V(5,1)?
V(6,2) = 30. V(5,1) = 5. One seat from five is simply n.
What is V(n,0)?
1. One empty list. The denominator is n! / n! = 1, through 0! in the (n-0)! writing.
Does the calculator round a fractional n?
It takes the integer part of n and k. 5.8 and 3.2 count as V(5,3) = 60.
Why subtract k in the denominator?
(n-k)! is the factorial of the people you do not seat. You divide by the orders you never arrange.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.