Permutations vs Combinations

The Idea

Three podium places, five runners: five choices, then four, then three. That is 60 ordered line-ups — a permutation, n! / (n - k)!.

If only the group matters, each trio has been counted 3! times, once per ordering. Divide that away and 60 becomes 10. Pascal's triangle stores the same counts: every entry is the two above it, because the newest runner is either in or out.

Real-World Example

A/B testing asks combinations: how many pairs of variants can be compared. Scheduling asks permutations: how many orders the same tasks could run in. Reaching for the wrong one is why an estimate comes back six times too large.

The Code

from math import comb, factorial, perm

print(perm(5, 3))                    # 60  -> ordered podiums
print(factorial(5) // factorial(2))  # 60  -> the same n! / (n-k)!
print(comb(5, 3))                    # 10  -> order thrown away
print(perm(5, 3) // factorial(3))    # 10  -> divided by k! by hand
print(comb(5, 3) == comb(5, 2))      # True -> pick who is left out instead

# turn ordered picks into unordered groups
combinations = perm(n, k) // ___