Permutations Versus Combinations

Seat 3 students in a row and swapping two of them makes a new seating. Pick 3 students for a committee and swapping two names changes nothing — it's the same committee. That's the whole difference. Permutations count outcomes where rearranging matters; combinations count outcomes where it doesn't.

The link between them is division. If you pick 3 people from 10 in order, you get 10 x 9 x 8 = 720 ordered lists. But each group of 3 people shows up 6 times in that list, once for every way to shuffle those 3 (3! = 3x2x1 = 6). So the number of unordered groups is 720/6 = 120. That division by the group's own internal orderings is what turns a permutation count into a combination count.

Before reaching for a formula, ask one question: if I swap two selected items, do I get a different outcome? If yes — places, prizes, rankings, seats — it's a permutation. If no — a committee, a team, a handful of toppings — it's a combination. This same ordered-versus-unordered split shows up again when counting outcomes for probability: whether an event counts arrangements or just groups changes the denominator you build your probability from, so getting this choice right here protects every calculation downstream.