Statistics Combinations

What Is a Combination?

A combination is a selection of items where order does not matter. Choosing 3 books out of 5 to pack for a trip is a combination problem, because packing book A, then B, then C ends up being the exact same trio as packing C, then A, then B — you only care which books end up in the bag, not the order you grabbed them.

The Combination Formula

The number of ways to choose r items from n, when order doesn't matter, is C(n, r) = n! / (r! (n − r)!). Combinations are related to permutations by dividing out the r! ways to order each group: C(n, r) = P(n, r) / r!, since every group of r items would otherwise be counted once for each of its possible orderings.

import math

n = 5  # books available
r = 3  # books chosen for the trip

combinations = math.comb(n, r)
permutations = math.perm(n, r)

print(f"C(5,3) = {combinations}")  # 10
print(f"Check via nPr / r! = {permutations // math.factorial(r)}")  # 10

With 5 books labeled A through E, choosing 3 to pack gives exactly 10 distinct groups. Order never matters here — 'A, B, C' and 'C, A, B' describe the same trio, so both collapse into a single combination.

1. ABC
2. ABD
3. ABE
4. ACD
5. ACE
6. ADE
7. BCD
8. BCE
9. BDE
10. CDE

Combinations vs Permutations

import math

lottery_combinations = math.comb(49, 6)
print(f"Ways to choose 6 numbers out of 49: {lottery_combinations:,}")
# 13,983,816 — this is exactly why winning a 6/49 lottery is so unlikely!

Combinations are the right tool whenever you're forming a subset rather than a sequence — picking a committee from a pool of candidates, dealing a poker hand, or choosing which toppings go on a pizza, since the final group is what matters, not the order items were added.