When Order Doesn't Matter

Combinatorics · Axiom Academy

INTRO When Order Doesn't Matter Discover how combinations differ from permutations when arranging the same elements. Same People, Different Questions We're choosing 3 people from 5 friends: Alice, Bob, Carol, Dan, and Eve. Watch what happens when we ask different questions. All Possible Officer Elections Let's list every way to elect a President, VP, and Treasurer from Alice, Bob, and Carol. Each arrangement is different because the roles are different. Now imagine there are no specific roles - just a committee of 3. Watch what happens to our 6 arrangements... This pattern reveals the fundamental connection between permutations and combinations. P(3,3) = 3! = 6 r! = 3! = 6 C(3,3) = 6 ÷ 6 = 1 Each group of r items can be arranged r! ways. We divide to eliminate these duplicates. Permutations: Order matters (positions, rankings, sequences) Combinations: Order doesn't matter (teams, committees, selections) Combinations are always ≤ permutations because we're grouping together arrangements that differ only in order. C(n,r) = P(n,r) ÷ r! = n! ÷ (r! × (n-r)!)

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