• UNIT 15 : Combinatorics

    Key unit competence 

    Use combinations and permutations to determine the number of ways a random experiment occurs. 

    Learning objectives 



    15.1 Counting techniques 

    Combinatorics , also known as combinatorial analysis, is the area of mathematics concerned with counting strategies to calculate the ways  in which objects can be  arranged to satisfy given conditions. 

    Venn diagrams 

    This is a way of representing sets in a closed curve. They are usually circular or oval shaped. The universal set, written U or , is represented by a rectangle and is a set containing all the sub-sets considered in the problem. In the probability 

    theory, the universal set is called the sample space and its subsets are called the events. 



    Tree diagrams 

    A tree diagram is a diagram with a structure of branching connecting lines representing a  relationship. It can be used to find the number of possible outcomes of experiments where each experiment occurs in a finite number of ways. For example, when you toss a coin, the outcome is either head or tail. A second toss would also give head or tail. We represent this as:



    Contingency table

    This is a method of presenting the frequencies of the outcomes from an experiment in which the observations in the sample are categorised according to two criteria. Each cell of the table gives the number of occurrences for a particular combination of categories. An example of contingency table in which individuals are categorised by gender and performance is shown below. 


    A final column giving the row sums and a final row giving the column sums may be added. Then the sum of the final column and the sum of the final row both equal the number in the sample. 

    If the sample is categorized according to three or more criteria, the information can be presented similarly in a number of such tables. 


    Multiplication rule 





    15.2 Arrangements and permutations

    Mental task

    Imagine that you are a photographer. You want to take a photograph of a group of say 7 people. In how many different ways can they be arranged in a single row?

    Factorial notation 



    Permutations

    Activity 15.1 

    Discuss in groups of 5: in how many different ways can a group of 3  students be arranged to sit in a row? 

    Share your findings with the rest of the class. 


    Arrangements without repetition 

    Suppose that two of the three different pictures, A, B and C are to be hung, in line, on a wall of three places. The pictures can be hung in different orders: 


    Each of these orders is a particular arrangement of the pictures and is called a permutation.





    Arrangements with repetition 




    Arrangement in a circle 

    Observe the arrangement of four letters A, B, C and D on a circle as  shown below



    Conditional arrangements 

    Sometimes arrangements may have certain restrictions which should be dealt with first.



    Arrangement with indistinguishable elements 

    In arranging objects, some sets may contain certain elements that are indistinguishable from each other. Below is how we find the number of arrangements. 





    15.3 Combinations 

    Activity 15.2 

    Discuss in groups of 5: in how many ways can a committee of 5 students be chosen from a class of 30 students? 

    Share your findings with the rest of the class. 

    Definitions and properties 




    Conditional combination 

    Sometimes we are given a condition that must be taken into account in a combination. 








    Pascal’s triangles   






    Binomial expansion 









    UNIT 14 : Measures of dispersionUNIT 16 : Elementary probability