Combinatorics — Counting, Permutations, Combinations

GRE Math Subject · Axiom Academy

LESSON Combinatorics -- Counting, Permutations, Combinations Fundamental counting principles, binomial coefficients, and classic GRE combinatorics Multiplication Principle: If task A can be done in ways and task B in ways, then A followed by B can be done in ways. Addition Principle: If task A can be done in ways and task B in ways (with no overlap), then A or B can be done in ways. Example: How many 3-letter strings can be formed from the 26-letter alphabet if repetition is allowed? A permutation is an ordered arrangement of objects. The number of ways to arrange objects chosen from distinct objects is: Special case: (arrange all objects) GRE problem: In how many ways can a president, vice president, and treasurer be chosen from a club of 10 members? Order matters (different offices), no repetition: Permutations with repetition: The number of ways to arrange objects where are identical of type 1, of type 2, etc.: Example: Arrangements of MISSISSIPPI: A combination is an unordered selection. The number of ways to choose objects from is: (total subsets of an -element set) GRE problem: A committee of 4 is chosen from 6 men and 5 women. How many committees have exactly 2 women? GRE problem: What is the coefficient of in ? Harder variant: Find the coefficient of in . Inclusion-Exclusion Principle (two sets): GRE problem: How many integers from 1 to 1000 are divisible by 3 or 5?

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