24/12/20211MATH221Mathematics for ComputerScienceUnit 8Introduction to Combinatorics2Learning Objectives•Introduce Multiplication and addition Rules: these form thebasis for work on combinatorics.•Introduce permutation and combination.3•Combinatoricsis an area that primarily concerning withcounting that is very important in Statistics especially in thearea of probability.•This unit will study the basic model for counting.•To use this model, we must model the problem according to themodel.Introduction4Theorem 8.1 The Multiplication RuleIf an operation consists ofksteps andthe first step can be performed inn1ways,the second step can be performed inn2ways(regardless of how the first step was performed),:thekthstep can be performed innkways(regardless of how the preceding steps were performed),Then the entire operation can be performed inn1n2n3…nkways.The Multiplication Rule

24/12/202125A typical PIN is a sequence ofany four symbolschosenfrom the26 lettersin the alphabet and the ten digits,with repetition allowed. Examples: CARE, 3387, B32B,and so forth.How many different PINs are possible?Example 1: No. of Personal Identification Numbers (PINs)You can think of forminga PIN as afour-stepoperationto fill in eachof the four symbols insequence.The Multiplication Rule6Step 1: Choose the first symbol.Step 2: Choose the second symbol.Step 3: Choose the third symbol.Step 4: Choose the fourth symbol.Example 1: No. of Personal Identification Numbers (PINs)There is a fixednumber of ways toperform each step,namely 36,regardless howpreceding stepswere performed.Hence, by themultiplication rule,there are:36363636 = 364=1,679,616PINs in all.7Example 2: No. of PINs without RepetitionNow, suppose thatrepetition is not allowed.How many different PINs are there?Step 1: Choose the first symbol.Step 2: Choose the second symbol.Step 3: Choose the third symbol.Step 4: Choose the fourth symbol.36 ways35 ways34 ways33 waysHence, by themultiplication rule, there are:36353433 =1,413,720PINs in all withno repeated symbol.8When the Multiplication Rule is Difficult/Impossible to ApplyExample 3:Consider the following problem:Three officers – a president, a treasurer, and a secretary – are tobe chosen from among four people: Ann, Bob, Cyd, and Dan.Suppose that, for various reasons,Ann cannot be presidentandeitherCyd or Dan must be secretary. How many ways can theofficers be chosen?It is natural to try to solve this problem using the multiplicationrule. A person might answer as follows:There are three choices for president (all except Ann), threechoices for treasurer (all except the one chosen as president),and two choices for secretary (Cyd or Dan).Therefore, by themultiplication rule, 332 =18Is this correct?

24/12/202139It isincorrect. The number of ways to choose thesecretary variesdepending on who is chosen forpresident and treasurer.

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