Pick 3 friends from 10 to share a pizza. Pick 3 to stand 1st, 2nd, 3rd in a race. Same numbers — but the second has 6× more answers. Order is everything.
Combinations count selections where order *doesn't* matter; permutations count arrangements where it *does*. Get the distinction right and the formula follows.
Lottery odds, password strength, poker hands, committee picks, scheduling, genetics — counting possibilities underpins probability itself.
How many ways to choose 3 people from 5? (order doesn't matter)
Permutations count order; combinations divide that by r! because the r chosen items can be arranged r! ways that all count as the same selection.
A pizza place has 8 toppings. How many 3-topping pizzas are possible?
How many ways can 1st, 2nd, 3rd be awarded among 8 runners?
Order matters → permutation. P(8,3) = 8 × 7 × 6 = 336 podiums.
'Combination lock' is misnamed — order matters on a real lock, so it's actually a *permutation* lock. In maths, combination = order-free.
C(n, r) values are exactly the entries of Pascal's triangle — row n, position r. The triangle is a combinations lookup table.
- Order matters → permutation P(n,r) = n!/(n−r)!.
- Order doesn't → combination C(n,r) = n!/(r!(n−r)!).
- C(n,r) = Pascal's-triangle entries — and the building blocks of probability.