University:
Georgia Institute of Technology
Course:
MATH 3215 | Introduction to Probability and Statistics
Academic year:

2024
Views:

267

Pages:

1
Author:

RonoEnduh6sl

Counting Principles and Examples • Fundamental Principle of Counting If one thing can be done in m ways and another thing can be done in n ways, the two things can be done in mn ways. o Example: A restaurant has 5 appetizers, 8 beverages, 9 entrees, and 6 desserts on the menu. If you have a beverage and a dessert, there are 8*6=48 different meals consisting of a beverage and dessert. o Example: In the last example suppose that you make a meal consisting of an appetizer, an entree, a dessert, and a beverage. Then there are 5*9*6*8=2160 different meals. o California vehicle license plates take the form DLLLDDD where the first D is any digit except 0, the other D's are any digit (including 0), and the L's are any letter of the alphabet. There are then 9*26*26*26*10*10*10=158,184,000 possible California vehicle licenses. • Permutations Permutations are arrangements of objects in which the order of objects in an arrangement is taken into account. o Example: Three people, called a, b, and c sit in three chairs arranged in a row. If a sits in the first chair, b in the second chair, and c in the third chair that is one possible arrangement. This arrangement can be written as abc. The other five arrangements are acb, bac, bca, cab, and cba. There are 6 different arrangements of these three people. o Example: Three people, again called a, b, and c sit in two chairs arranged in a row. Since there are only two chairs, only two of the people can sit at the same time. The arrangements are then ab, ba, ac, ca, bc, and cb. It can be said that there are 6 arrangements or permutations of 3 people taken two at a time. o The number of permutations of n things taken k at a time is nPk=n!/(n-k)! • Combinations Combinations are arrangements of objects in which the order of objects in an arrangement is not taken into account. o Example: Three people, a, b, and c, are available to work on a project that requires two workers. How many 2-person teams can be chosen for the project? There are only 3 2-person teams, a team consisting of a and b, a team comprised of a and c, and a team with b and c on it. Notice that a team consisting of b and a is the same as the team consisting of a and b. The order in which members of a team is written is not taken into account. o You have four cards numbered 1, 2, 3, and 4. How many 2-card hands can your deal from these four cards. The hands are 12, 13, 14, 23, 24, and 34. The order in which the cards are dealt is not important, it is only important to know which cards have ended up in a deal. o The number of combinations of n things taken k at a time is nCk=n!/[(n-k)!k!]

Counting Principles and Examples

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