Fundamental Counting Principle
The Fundamental Counting Principle is a concept in combinatorics that states that if there are m ways to do one thing and n ways to do another, then there are m * n ways to do both things. This principle is used to calculate the total number of possible outcomes when making a series of choices or decisions.
3 Key excerpts on "Fundamental Counting Principle"
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 7 Probability and Statistics CHAPTER 7 PROBABILITY AND STATISTICS The first part of this chapter discusses counting principles and how they relate to permutations and combinations, which are the building blocks of probability. The remainder of the chapter addresses statistics, introduces data measurements—such as mean, median, mode, and standard deviation—and reviews data analysis of various charts and graphs. THE Fundamental Counting Principle The Fundamental Counting Principle deals with identifying the number of outcomes of a given experiment and encompasses the counting rule : If one experiment can be performed in m ways, and a second experiment can be performed in n ways, then there are m × n distinct ways both experiments can be performed in this specified order. The counting principle can be applied to more than two experiments. PROBLEM A new line of children’s clothing is color-coordinated. Niki’s mother bought her five tops, three shorts, and two pairs of shoes for the summer. How many different outfits can Niki choose from? SOLUTION How many choices are there for a top? 5 How many choices are there for shorts? 3 How many choices are there for shoes? 2 Apply the counting principle: 5 × 3 × 2 or 30. Thus, Niki has 30 different outfits. PERMUTATIONS A permutation is an arrangement of specific objects in which order is of particular importance. To determine the number of possible permutations, the following for mula can be used: where n is the number of objects in the given set, r is the number of objects being chosen, and ! is the notation used for factorial. The factorial of a number is the product of that number and all the numbers less than it down to 1, or n ! = n × (n – 1) × (n – 2) × … × 3 × 2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The formula for permutations is a consequence of the counting rule described above. Example: Note that 4! cancels out part of 6! and we are left with only 6 × 5 = 30...
eBook - ePub- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
...In most cases the number will be too large to be enumerated completely. In such situations, often simple counting rules may be employed to systematically determine the total number of outcomes that are favorable to the event in question. The multiplicative rule of counting is one of the most basic tools in this respect. This rule and its generalizations are extremely useful in solving a large class of probability problems. The usefulness of these rules stems from the fact that, on many occasions, a statistical experiment is performed in different stages, and the entire experiment is a combination of these different parts. Definition 5.1. The multiplicative rule (or the basic principle) of counting states that, if task 1 can be performed in m ways, and task 2 can be performed in n ways, together they can be performed in m × n ways. Example 5.3. A soap manufacturer produces soaps in three different shapes (rectangular, round and oval) and in five different colors (white, cream, pink, violet and green). How many different shape-color combinations does the manufacturer produce? The multiplicative rule of counting says that, since there are three shapes and five colors, 3 × 5 = 15 shape-color combinations are possible in this case. Definition 5.2. The above principle can easily be extended to the generalized multiplicative rule (generalized basic principle) of counting, which says that, if task 1 can be performed in n 1 ways, task 2 in n 2 ways, …, task k in n k ways, then the k tasks can be performed together in n 1 × n 2 × ··· × n k ways. Example 5.4. Continuing with the soap manufacturer example (Example 5.3), suppose that the manufacturer, apart from producing the soaps in three different shapes and five different colors, also uses three different fragrances (rose, sandalwood and lavender)...
eBook - ePub- Derek L. Waller(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...They give precise answers to many basic design situations and are similar to probability rules as they have a defined mathematical framework. In a sense they are a priori since required information is known before an experiment is performed. The difference with this counting theory is that probabilities are not involved. However, the counting rules can serve as the basis for probability. Outcomes from counting theory are deterministic. Single type of event With a single type of event the number of events is k, and the number of trials or experiments is n, then Total possible outcomes = k n 3(xiii) Suppose that a coin is tossed four times, then the number of trials, n, is four and the number of events, k, is two since heads or tails are the only two possible events. The events, obtaining heads or tails, are mutually exclusive since only heads or tails can be obtained in one throw of a coin. The collectively exhaustive outcomes are 2 4, or 16, as shown in Table 3.14 (H = Heads, T = Tails). Table 3.14 Different types of events Another rule involves different types of events. Here there are k 1 possible events on the first trial or experiment, k 2 possible events on the second trial, k 3 possible events on the third trial, and k n possible events on the n th trial. In this case the total possible outcomes of different events is calculated by the following relationship: Total possible outcomes = k 1 *k 2 *k 3 ………k n 3(xiv) As an illustration, my first car when I was a student in Newcastle on Tyne, England was an old Austin A40 with the license plate number of 212TPV. Thus, at that the time the pattern of the licenses issued by the Driving and Vehicle Center was 123ABC or three numbers followed by three letters. For numbers, there are ten possible outcomes; the numbers from 0 to 9. For letters there are 26 possible outcomes; from A to Z...
