Slide 1

Counting Rules

Tree Diagram

Addition principleTwo operations can be performed in A or B ways respectivelythen both operations can be performed together in A + B ways

Illustrative example 1To reach the top of the hill, Jack and Jill can use public transport (tram or bus) or private transport (hire-car, taxi or motorbike). In how many ways can Jack and Jill go up thehill if only one form of transport is to be used for the entire trip? Answer: 2 + 3 = 5 ways

Illustrative example 2A particular mathematics problem can be solved in 2 ways using analytical methods, in 4 ways using approximation techniques and in 3 ways by trial and error strategies. In how many ways can the problem be solved? Answer: 2 + 4 + 3 = 9 ways

Multiplication PrincipleIf two operations can be performed in A and B ways respectively, then both operations can be performed in succession in A B ways

Illustrative exampleFrom a cafeteria 4-course lunch menu, I can choose 3 varieties of soup, 5 types of seafood, 4 kinds of side dish and 2 types of salad.How many different dishes are offered?How many different lunches can be ordered if one dish from each course is selected?c.How many different types of dish are possible if soup and seafood must be included with each order?

Solutiona. The 4-course menu offers:3 + 5 + 4 + 2 = 14 different dishes.b.

Solutionc. soup and seafood only

soup and seafood and a sidedish only

Permutations involving restrictionsIdentical ObjectsGrouped objectsArrangements in a circle10Identical Objects The number of ways of arranging n objects, which include n1 identical objects of one type, n2 identical objects of another type, n3 identical objects of yet another type and so on is:

Grouped Objects If n objects are to be divided into m groups with each group havingG1, G2, G3, . . . Gm objects respectively,

the number of arrangements is given bym! G1! G2! G3! . . . Gm!

Illustrative example 1In how many ways can the letters A, B, C, D be positioned in a row if A and B must be next to each other?

Grouped AB or BA

Solution The 3 objects can be arranged in 3! ways and within the group A and B can themselves be arranged in 2! ways (namely AB and BA). The multiplication principle is now used so that the number of arrangements when A and B are together is 3! 2! = 12.

If A and B are to be together we consider the problem to be one of arranging 3 objects, say X, C and D, where one of the objects, X is the group containing A and B.Consider the permutations if A, B, C must be together.

Illustrative example 2Solution We view the letters as consisting of two objects X and D, where X is the group of letters A, B and C.Thus we have two objects to arrange in 2! WaysXD and DX

Illustrative Example 3 Five cars a Toyota, a Ford, a Commodore, a Corolla and a BMW are to be parked side by side. In how many ways can this be done if the Toyota and BMW are not to be parked next to each other?

SolutionThe five cars can be arranged in 5! ways withoutrestriction. Number of ways of arranging 5 cars = 5!Calculate the number of arrangements where the Toyota and BMW are together (4! 2!).Number of ways where the Toyota and BMW are not together = 5! 4! 2!(Subtract from the unrestricted number of arrangements the number of ways the two cars are together)

Illustrative Example 4 The letters of the word REPLETE are arranged in a row. In how many ways can this be done if the letters R and P must not be together?SolutionFind the number of unrestricted arrangements of the 7 letters and consider that there are 3 identical Es.Calculate the (restricted) number of ways R and P are together. Consider R and P as one object so there are 6 objects to arrange. There are three Es to consider (3! ways). R and P can be arranged in 2! ways within their group.Subtract the number of ways with R and P together from the total number of arrangements.

Arrangements in a circle n distinguishable objects can be arranged in a circle in

(n 1)! ways.

Illustrative exampleAnna, Betty and Lin stand on the circumference of a circle painted on the schools playground. How many different arrangements are there?Solution

Notice that Anna is locked in position to provide a reference point and Betty and Lin are arranged around Anna in 2! (= 2) ways.Illustrative Example 2Continuing previous problem Susie now joins the group to make 4 people in a circle.

How many different arrangements are now there?Solution

We can designate any of the 4 girls in the circle as our start by fixing one person (in this case, Anna) in one position and arranging the remaining girls around her. This reduces by one person the number of girls to arrange.Illustrative example 1A captain and vice-captain are to be chosen from a group consisting of 10 cricket players. From the remaining 8 players, 3 will be selected to be the wicket keeper, spin bowler and fast bowler. Calculate how many different ways the 5 positions can be allocated.

A box contains 16 marbles numbered 1, 2, 3, . . . , 16. One marble is randomly selected.Let A be the event the marble selected is a prime number greater than 3 and let B be the event the marble selected is an odd number.

Are A and B mutually exclusive events (MEE) or Independent Events (IE)?

Illustrative ExampleA card is chosen from a pack of 52 playing cards and its suit noted; then it is returned to the pack before another card is chosen.Is this MEE or IE

Calculate the probability of choosingi. two hearts ii. a diamond then a spade iii. a heart and a club.