Basic Counting Rule; Permutations; Combinations Lecture 4 Basic Counting Rule; Permutations; Combinations Basic Counting Rules Text: A Course in Probability by Weiss 3:1 ∼ 3:3 Permutations Combinations STAT 225 Introduction to Probability Models January 20, 2014 Whitney Huang Purdue University 4.1 Basic Counting Rule; Agenda Permutations; Combinations 1 Basic Counting Rules Basic Counting Rules Permutations Combinations 2 Permutations 3 Combinations 4.2 Basic Counting Rule; Basic Counting Rule Permutations; Combinations The Basic Counting Rule is used for scenarios that have multiple choices or actions to be determined. Basic Counting Rules Permutations Combinations 4.3 Basic Counting Rule; Factorial Permutations; Combinations Definition The factorial of a non–negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Example: Basic Counting Rules 5! = 5 × 4 × 3 × 2 × 1 = 120 Permutations Combinations Convention: 0! = 1 4.4 Basic Counting Rule; Permutation Permutations; Combinations Definition A permutation of r objects from a collection of n objects is any ordered arrangement of r distinct objects from the n objects. Basic Counting Rules Notation: (n)r or nPr Permutations n! Formula: (n)r = (n−r)! Combinations The special permutation rule states that anything permute itself is equivalent to itself factorial. Example: 3! 3! (3) = = = 3! = 3 × 2 × 1 = 6 3 (3 − 3)! 0! 4.5 Basic Counting Rule; Permutation Permutations; Combinations Basic Counting Rules Permutations Combinations 4.6 Basic Counting Rule; Combination Permutations; Combinations Definition A combination of r objects from a collection of n objects is any unordered arrangement of r distinct objects from the n total objects. Basic Counting Rules Remark: The difference between a combination and a Permutations permutation is that order of the objects is not important for a Combinations combination. n Notation: r or nCr n n! Formula: r = (n−r)!×r! 4.7 Basic Counting Rule; Combination Permutations; Combinations Basic Counting Rules Permutations Combinations 4.8 Basic Counting Rule; Ordered partition Permutations; Combinations Definition An ordered partition of m objects into k distinct groups of sizes m1; m2; ··· ; mk is any division of the m objects into a combination of m1 objects constituting the first group, m2 objects comprising the second group, etc. The number of such Basic Counting Rules partitions that can be made is denoted by m Permutations m1;m2;··· ;mk Combinations Remark: We called m the multinomial coefficient where m1;m2;··· ;mk m m! = m1; m2; ··· ; mk m1! × · · · mk ! 4.9 Basic Counting Rule; Example 13 Permutations; Combinations 3 people get into an elevator and choose to get off at one of the 10 remaining floors. Find the following probabilities: 1 P(they all get off on different floors) 2 P(they all get off on the 5th floor) Basic Counting Rules 3 P(they all get off on the same floor) Permutations Combinations 4 P(exactly one of them gets off on the 5th floor) 5 P(at least one of them gets off on the 5th floor) Solution. 1 (10)3 10×9×8 = 103 = 10×10×10 = 0:72 2 1 = 103 = 0:001 3 10 = 103 = 0:01 4 (3)1×1×9×9 = 103 = 0:243 3 5 9 = 1 − 103 = 0:271 4.10 Basic Counting Rule; Example 14 Permutations; Combinations Suppose we have the fictional word “DALDERFARG" 1 How many ways are there to arrange all of the letters? 2 What is the probability that the 1st letter is the same as the 2nd letter? Basic Counting Rules 3 What is the probability that an arrangement of all of the Permutations letters has the 2 D’s next to each other? Combinations 4 What is the probability that an arrangement of all of the letters has the 2 D’s next to each other and it has the 2 R’s grouped together (not necessarily the D’s and R’s next to each other)? 5 What is the probability that an arrangement of all the letters has the 2 D’s before the F? 4.11 Basic Counting Rule; Example 14 Permutations; Combinations Solution. 1 10 10! 2;2;2;1;1;1;1 = 2!×2!×2! Basic Counting Rules 2 (i) = 6 × 1 = 1 10 9 15 Permutations 3 × 8! (1) 2!×2! 2!×3 1 Combinations (ii) = 10! = 10×9 = 15 2!×2!×2! 9! 3 2!×2! 2! 1 = 10! = 10 = 5 2!×2!×2! 8! 4 2! 2!×2! 2 = 10! = 10×9 = 45 2!×2!×2! 5 Three possible ordering for “D", “D", “F" DDF DFD FDD 1 )= 3 4.12 Basic Counting Rule; Summary Permutations; Combinations In this lecture, we learned Basic counting rules Combinations and Permutations Basic Counting Rules Permutations Combinations 4.13.