Counting Methods — Permutations and Combinations

GRE Quantitative · Axiom Academy

LESSON Counting Methods — Permutations and Combinations Count arrangements and selections without listing every case — starting from one rule, and one picture of what "order" really changes. 1. The Fundamental Counting Principle If a first choice can be made in m ways and a second, independent choice can be made in n ways, the whole sequence of choices can be made in ways. Every formula in this lesson is just this rule applied over and over. 3 shirts, then 2 pairs of pants — every branch of the first choice re-opens all of the second 2. Permutations — Derived by Filling Slots Say you have 5 books and want to choose 3 to place on a shelf, in order (1st, 2nd, 3rd position). Fill the slots one at a time: the 1st slot has 5 candidates, but once a book is placed there, only 4 remain for the 2nd slot, and only 3 for the 3rd. Order matters here — swapping which book sits 1st vs. 2nd gives a genuinely different shelf. 5 × 4 × 3 — three shrinking pools, multiplied by the Counting Principle the same product, written compactly with factorials 3. The Same 3 Books, Every Order — Then One Group Here's the picture that separates the two ideas. Pick the same trio of books — say — out of the 5 on the shelf. As permutations , that trio can sit on the shelf in 3! = 6 distinct orders — watch each one play out below. But if all you're doing is choosing which 3 books make the trip (a combination), every one of those 6 arrangements is the same choice — they collapse into a single circled group.

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