The Art of Choosing

Combinatorics · Axiom Academy

Discover the power of combinations: when order doesn't matter, everything changes. Let's start with three letters: A, B, and C. Click the buttons below to see something interesting... Different orders = different outcomes Order doesn't matter, just the selection You need to choose 2 people from a group of 4 for a project. Pick any 2 people and see what happens... Choose 3 toppings for your pizza from the 5 options below. Click toppings to add or remove them! Every combination problem asks the same core question in different disguises... How many ways to choose 2 people from 4? How many ways to choose 3 toppings from 5? How many ways to choose 5 cards from 52? How many ways to choose 3 books from 10? Combinations count selections where order doesn't matter. A,B,C is the same as C,B,A . This makes combinations fundamentally different from permutations (arrangements). Combinations appear everywhere: forming teams, selecting committee members, choosing lottery numbers, dealing card hands, picking pizza toppings, and selecting items for gift bags. There's a powerful formula that answers "how many ways can I choose r objects from n objects?" where n! means "n factorial" = n × (n-1) × (n-2) × ... × 2 × 1

This is the written version of the interactive lesson above. See the full Combinatorics course.