1st Year Math Notes Chapter 7

• Our comprehensive 1st Year Math Notes Chapter 7 will ensure you're fully prepared for your exams.
• What is the factorial with its notation and examples
  
• Definition of permutation with notations and fundamental principle of counting
  
• What is the circular permutation with examples
• What is a combination of examples and notations
• What is a complementary combination with examples
• What is a complementary combination with examples
• Mutually exclusive disjoint events with examples
• Equally likely events with examples
• Estimating Probability and Tally marks
• Multiplication of probability is dependent event independent events
• The factorial notation was introduced by Christian Kramp (1760 - 1826) in 1808. This notation will be frequently used in this chapter as well as in inding the Binomial Coeicients in later chapter. Let us have an introduction of factorial notation. Let n be a positive integer. Then the product n(n - 1)(n - 2). . . . 3 . 2 . 1 is denoted by n! or In and read as n factorial.
• Thus for a positive integer n, we deine n factorial as: n! =n(n - 1)! where 0!= 1
• Permutation: Suppose we like to ind the number of diferent ways to name the triangle with vertices A, B and C. The various possible ways are obtained by constructing a tree diagram as follows:
• To determine the possible ways, we count the paths of the tree, beginning from the start to the end of each branch. So, we get 6 diferent names of triangle. ABC, ACB, BCA, BAC, CAB, CBA. Thus there are six possible ways to write the name of the triangle with vertices A, B and C. Explanation: In the igure, we can write any one of the three vertices A, B, C at irst place. After writing at irst place any one of the three vertices, two vertices are left. So, there are two choices to write at second place. After writing the vertices at two places, there is just one vertex left. So, we can write only one vertex at third place.
• Fundamental Principle of Counting: Suppose A and B are two events. The irst event A can occur in p diferent ways. After A has occurred, B can occur in q diferent ways. The number of ways that the two events can occur is the product p.q.
• This principle can be extended to three or more events. For instance, the number of ways that three events A, B and C can occur is the product p.q.r. One important application of the Fundamental Principle of Counting is to determine the number of ways that n objects can be arranged in order. An ordering (arrangement) of n objects is called a permutation of the objects. A permutation of n diferent objects is an ordering (arrangement) of the objects such that one object is irst, one is second, one is third and so on. According to Fundamental Principle of Counting:
• i) Three books can be arranged in a row taken all at a time = 3.2.1 = 3! ways
• ii) Number of ways of writing the letters of the WORD taken all at a time = 4.3.2.1 = 4!
• Each arrangement is called a permutation. Now we have the following deinition. A permutation of n diferent objects taken r (7 n) at a time is an arrangement of the r objects. Generally it is denoted by nPr or P(n,r).
• n diferent objects can be arranged taken all at a time in n! ways.