Slide 1

Slide 2 PERMUTATIONS AND COMBINATIONS Slide 3 Fundamental Counting Principle If there are n(A) ways in which an event A can occur, and if there are n(B) ways in which a second event B can occur after the first event has occurred, then the two events can occur in n(A) n(B) ways. Special products such as 4! (4 3 2 1) (or any other constant) frequently occur in counting theory. This symbol is a special notation, known as factorial. Factorial is explained as: For any positive integer n, we define n-factorial, written as n! = n(n - 1)(n - 2)(n - 3).. We define 0! =1 Slide 4 Permutation and Combination Formulas Permutation - The number of possible distinct arrangements of r objects chosen from a set of n objects is called the number of permutations of n objects taken r at a time and it equals: nPr = __n!__ (n r)! Slide 5 Permutation and Combination Formulas Example In how many ways can a president, vice president, secretary, and treasurer be selected from an organization with 20 members? Solution (the number of arrangements in which 4 people can be selected from a group of 20) n = 20 r = 4 n P r = 20!__ = 20 19 18 17 16! = 116,280 (20 - 4)! 16! Slide 6 Permutation and Combination Formulas Combination - The number of combinations of n objects taken r at a time is: nCr = ___n!___ r!(n r)! Slide 7 Permutation and Combination Formulas Example In the Texas lottery you choose 6 numbers from 1 though 54. If there is no replacement or repetition of numbers, how many different combinations can you make? Solution n = 54 r = 6 n C r = 54!__ = 54 53 52 51 50 49 = 25,827,165 6! (54-6)! 720 Slide 8 Permutations and Combinations Links Probability Handout Probability Handout Probability Workshop Probability Workshop