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Delve into discrete versus continuous counting, permutations, combinations, and subsets with practical examples to grasp the fundamentals of combinatorics effortlessly. Learn the essential formulas for efficiently counting large sets. Discover how to solve various counting problems through clear explanations and illustrations.
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9.1 Basic Combinatorics
What you’ll learn about • Discrete Versus Continuous • The Importance of Counting • The Multiplication Principle of Counting • Permutations • Combinations • Subsets of an n-Set … and why Counting large sets is easy if you know the correct formula.
Example Arranging Three Objects in Order How many three-letter codes can be formed using A, B, C, and D if no letter can be repeated?
Example Arranging Three Objects in Order How many three-letter codes can be formed using A, B, C, and D if no letter can be repeated? List the possibilities in an orderly manner: So there are 4 • 6 = 24 possibilities.
Example Arranging Three Objects in Order A tree diagram can be used to obtain the same result.
Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.
Example Using the Multiplication Principle If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used. You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.
Permutations of an n-Set There are n! permutations of an n-set.
Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.
Example Distinguishable Permutations Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER. Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.
Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?
Example Counting Combinations How many 10 person committees can be formed from a group of 20 people?
