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Combinatorics

# Combinatorics - Outline The Product Rules for Counting...

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Outline The Product Rules for Counting Permutations Combinations Combinatorics Michael Akritas Michael Akritas Combinatorics

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Outline The Product Rules for Counting Permutations Combinations The Product Rules for Counting Permutations Permutations of Distinct Units Permutations When Some Units are Identical Combinations Applications in Probability Combinations-Related Results Michael Akritas Combinatorics
Outline The Product Rules for Counting Permutations Combinations Why Count? 1. How many 7-place license plates are possible if the first 3 places are letters and the last 4 are numbers? 2. How many groups of 5 cards can be formed from a deck of 52 cards? 3. If each of 10 people in a room shakes hands with everybody else how many handshakes take place? 4. How many n -long sequences consisting of k 1s and n - k 0s can be formed? 5. How many paths going from the lower left corner of a 4 × 3 grid to its upper right corner are there? 6. How many linear arrangements of m defective and n functional antennas in which no two defectives are consecutive are there? The mathematical theory of counting is known as combinatorial analysis . Michael Akritas Combinatorics

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Outline The Product Rules for Counting Permutations Combinations The Simple Product Rule: Suppose a task can be completed in two stages. If stage 1 has n 1 outcomes, and if stage 2 has n 2 outcomes regardless of the outcome in stage 1, then the task has n 1 × n 2 possible outcomes. Example 1. In how many ways can we select the 1st and 2nd place winners from the four finalists Niki, George, Sophia and Martha? 2. Ten women, each of which has three children make up the finalists in the ”mother and child of the year” award. In how many ways can the winning mother-child pair be selected? ANSWER: 1. 4 × 3 = 12. 2. 10 × 3 = 30. Michael Akritas Combinatorics
Outline The Product Rules for Counting Permutations Combinations The General Product Rule: If a task can be completed in k stages and stage i has n i outcomes, regardless of the outcomes the previous stages, then the task has n 1 × n 2 × · · · × n k outcomes. Example 1. In how many ways can we select a 1st, 2nd and 3rd place winners from Niki, George, Sophia and Martha? 2. A college planning committee consists of 3 freshmen, 4 sophomores, 5 juniors and 2 seniors. A subcommittee of 4, consisting of 1 person from each class is to be chosen. How many different subcommittees are possible? 3. How many functions defined on n points with range { 0 , 1 } are possible? ANSWERS: 1. 4 × 3 × 2 = 24. 2. 3 × 4 × 5 × 2 = 120. 3. 2 n Michael Akritas Combinatorics

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Outline The Product Rules for Counting Permutations Combinations Permutations of Distinct Units Permutations When Some Units are Identical The answer to the first question in previous slide is called the number of permutations of 3 subjects selected from 4 subjects .
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