The rule
V(n, k) = n! ÷ (n − k)!
Why it works
A variation, or partial permutation, counts ordered selections. There are n choices for the first place, n−1 for the second, and so on, which is exactly n! divided by the factorial of what is left over.
How to do it by hand
- Count the choices for the first position: n
- For the second: n − 1, and so on for k positions
- Multiply them all together
- With repetition allowed it is simply n to the power k
What is worth knowing
The distinction from combinations is exactly a factor of k!, which is why order costs so much: choosing 3 from 10 gives 120 combinations but 720 variations. Variations with repetition are what password strength rests on, and the numbers are unforgiving in both directions: a 4-digit PIN is 10⁴, ten thousand possibilities, which a computer exhausts instantly. An 8-character password from 94 printable characters is 94⁸, about six quadrillion — still breakable, which is why length beats complexity.