Permutations Formula

Algebra 2 · Axiom Academy

LESSON The Permutations Formula Why arranging r of n items in order gives — built from one idea: order matters. A permutation is an arrangement in which the order is part of the outcome. Move the same items into different positions and you have a different permutation — ABC and CBA are not the same arrangement. Gold, Silver, Bronze from 5 runners. Swapping who gets gold and bronze is a different result. Picking a 3-person committee from 5. The same three people are the same committee. To fill r ordered positions, use the multiplication principle: multiply the number of choices at each step. Each pick uses up one item, so the number of choices counts down. The descending product n(n−1)⋯(n−r+1) is the start of a factorial that stops early. Write the full factorial on top, then divide by the factorial of exactly the factors you skipped. Now just substitute and evaluate. To arrange 3 of 5 books, take n = 5 and r = 3 , drop them into the formula, and reduce. You built the permutation formula from one idea — order matters — and saw why the factorial is the natural way to write it.

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