Section 8.3 PROBABILITY

FACTORIALS, COMBINATIONS AND PERMUTATION 8.3 COUNTING RULE FACTORIALS, COMBINATIONS AND PERMUTATION The task of determining final outcomes in an experiment could be daunting, especially if the experiment is repeated a large number of times. For this reason, we must find a better way. Example #24 Now, suppose a dice is rolled 4 times, what is the number of possible outcomes? Solution: The 1st die has 6 outcomes The 2nd die has 6 outcomes for each outcome in 1st die The 3rd die has 6 outcomes for each outcome in 2nd die The 4th die has 6 outcomes for each outcome in 3rd die Therefore, Total outcomes = 6.6.6.6 = 1296 2

Counting Rule General Statement Let an experiment be performed four times. The 1st experiment results in n outcomes, the 2nd results in m outcomes, the 3rd results in k outcomes, and the 4th results in l outcomes. Then, Total number of outcomes = n × m × k × l Solution This experiment has two steps: Selection of an ice cream flavor Selection of a sundae topping Example #25 A small ice cream shop has 10 flavors of ice cream and 5 kinds of toppings for its sundaes. How many different selections of one flavor of ice cream and one kind of topping are possible? For ice cream flavor, the Number of outcomes = 10 For topping, the Number of outcome = 5 Therefore, the total number of possible outcomes = 10 × 5 = 50 3

FACTORIALS, COMBINATIONS, AND PERMUTATIONS Factorial is denoted by the symbol “!”. The factorial of a number is calculated by multiplying all integers from the number to 1. Formal Definition The symbol n!, is define as the product of all the integers from n to 1. In other words, n! = n(n - 1)(n – 2)(n – 3) · · · 3 · 2 · 1 Also note that by definition, 0! = 1 Example #26 4

Combinations Suppose we want to select two books from 6 books, and the following conditions applies: 1. This type of selection is without replacement. 2. The order in which selection is made is not important. That is, it is the same selection whether we select book 1 first (1,2) or book 2 first (2,1). Then, the question is how many ways can we make this selection? Of course, we can do it the old fashion way, write the selection as, (1,2) (1,3) (1,4) (1,5) (1,6) (2,3) (2,4) (2,5) (2,6) (3,4) (3,5) (3,6) (4,5) (4,6) (5,6) Therefore, we can select 2 books from 6 books 15 different ways. Note: Each of the possible selection is called a combination. 5

Combinations Definition Combinations give the number of ways x element can be selected from n distinct elements. The total number of combinations is given by, and is read as “the number of combinations of n elements selected x at a time.” The formula for the number of combinations for selecting x from n distinct elements is, Note: 6

Combinations Example #27 7

Permutations Permutation is similar to combination except that the order of selection is important. For example, how many ways can we select two letters from “A”, “B”, and “C” without replacement and if the order does not matter? This selection is a combination and the total number of combinations is These 3 ways are “AB”, “AC, and “BC”. Now suppose the order matters, then the two letter-combination can be selected or arranged in 6 ways: “AB”, “BA”, “AC”, “CA”, “BC” and “CB” These 6 selections are called 6 permutations or arrangements. 8

Permutations Example #28 Definition Permutations give the total selection of x elements from n different elements such that the order of selection matter. The notation and formula is Example #28 9

Permutations Example #29 Solution A ski patrol unit has nine members available for duty, and two of them are to be sent to rescue an injured skier. In how many ways can two of these nine members be selected? Now suppose the order of selection is important. How many arrangements are possible in this case? Solution 10