Permutations and Combinations – Learning Outcomes Count the arrangements of 𝑛 distinct objects. Count the number of ways of arranging 𝑟 objects from 𝑛 distinct objects. Count the number of ways of selecting 𝑟 objects from 𝑛 distinct objects. Compute binomial coefficients.

Recall Keywords When anything happens, we call it an event (or sometimes an experiment). The specific way the event ends is called the outcome of that event. The list of possible outcomes is called the sample space. e.g. rolling a die is an event. e.g. rolling a 4 on a die is an outcome. e.g. {1, 2, 3, 4, 5, 6} is the sample space of rolling a die.

Arrange 𝑛 Distinct Objects Recall the Fundamental Principle of Counting: If one event has 𝑚 outcomes and a second event has 𝑛 outcomes, then in total there are 𝑚×𝑛 outcomes for the pair of events. e.g. If you can choose four colours for each of three different models of car, how many different cars can you buy? e.g. How many meals can you order from a set menu if you must choose one starter (fruit salad, mozzarella sticks, or vegetable soup), one main course (steak, pasta, curry, or burger), and one dessert (ice cream or trifle).

Arrange 𝑛 Distinct Objects How many ways can you arrange your English, maths, and Irish textbooks on a shelf? Placing each book is an event. The possible books you can place each time is the sample space. The book you actually place each time is the outcome.

Arrange 𝑛 Distinct Objects For your first book, you have three options (English, maths, or Irish). Whichever book you choose first, you are left with two options for your second placement. When you have two books placed, you have only one choice left for the third book. Thus, there are 3×2×1=6 ways to arrange the books. In general, with no restrictions, there are 𝑛! (pronounced “𝑛 factorial”) =𝑛 𝑛−1 𝑛−2 …(3)(2)(1) ways to arrange 𝑛 distinct objects.

Arrange 𝑛 Distinct Objects e.g. In how many ways can the letters in the word ULSTER be arranged if: there are no restrictions? the words must begin with R? the words must begin with U and end with T? the words must begin with a vowel? the words must not begin with a vowel?

Arrange 𝑛 Distinct Objects e.g. Snow White is rearranging the dwarves’ bedroom. In how many ways can she arrange the beds if they must be in a different order to what they are now? e.g. If Doc needs to be beside the door, how many ways can the beds be arranged?

Arrange 𝑛 Distinct Objects Peter is arranging books on a shelf. He has five novels and three poetry books. He wants to keep the five novels together and the three poetry books together. In how many different ways can he arrange the books? 2014 (s) Q1

Arrange 𝑟 from 𝑛 Distinct Objects What if we have more choices than spots? e.g. a museum exhibit wants to display models of the Stargate alien races. It can choose from Alterans, Asgard, Furlings, Goa’uld, Jaffa, Nox, Replicators, Unas, and Wraith. There is only room for five displays however. In how many ways can five races be chosen and arranged in the display? These are called permutations. In general, there are 𝑛𝑃𝑟 ways of arranging 𝑟 out of 𝑛 objects (a button on your calculator).

Arrange 𝑟 from 𝑛 Distinct Objects Six horses are competing in a race. When making a bet, how many ways can you: pick the first place horse? pick the “place” horses (i.e. the first, second, and third place horses)?

Arrange 𝑟 from 𝑛 Distinct Objects A bank issues a unique six-digit PIN to each of its online customers. The password may contain any of the digits 0-9 in any position and the numbers may be repeated. How many different PINs are possible? How many different PINs do not contain a zero? How many different PINs contain at least one zero?

Arrange 𝑟 from 𝑛 Distinct Objects If Mr. Lawless implemented a seating plan, in how many ways could he arrange students in the front row of four seats? How many of these arrangements have Lorna and Lauren sitting next to each other?

Select 𝑟 from 𝑛 Distinct Objects In permutations, the arrangement of the objects is important. For combinations, the arrangement is unimportant – only the objects selected is important. e.g. Greta wants to make a sandwich and has the following ingredients in the fridge: mushroom, eggplant, tomato, and avocado. How many sandwiches can she make with two of these ingredients? The order of the ingredients in unimportant, so a sandwich with tomato and avocado is the same as a sandwich with avocado and tomato.

Select 𝑟 from 𝑛 Distinct Objects To get all the ways of arranging the sandwich: e.g. 4𝑃2=12 Some of these cases have the same ingredients in different order. To remove these extra cases, divide by 𝑟!. 4𝐶2= 12 2! =6 The remaining cases are called combinations and 𝑛𝐶𝑟 is pronounced “𝑛 choose 𝑟” This can also be written 𝑛 𝑟 .

Select 𝑟 from 𝑛 Distinct Objects In 2018, Boland, Durcan, Frost, Hopkins, Keats, Larkin, Montague, and Ní Chuilleanáin are prescribed poets on the leaving certificate English course. How many combinations of six poets are available to study?

Select 𝑟 from 𝑛 Distinct Objects There are 15 teams in a volleyball league. How many matches must be organised if: each team plays each other team once? each team plays each other team twice?

Select 𝑟 from 𝑛 Distinct Objects Seven men and six women are to be elected onto a committee of six people. How many ways can the members of the committee be elected if: there are no restrictions? Ms O’Brien was the chairperson last year and must remain on the committee? Mr Kelly has been found to have a conflict of interest and cannot be elected? the committee must have three men and three women.

Select 𝑟 from 𝑛 Distinct Objects How many triangles can be made by joining together three of the points on this circle? How many of these are equilateral?

Compute Binomial Coefficients Recall the binomial theorem: 𝑥+𝑦 𝑛 = 𝑟=0 𝑛 𝑛 𝑟 𝑥 𝑛−𝑟 𝑦 𝑟 Notice the 𝑛 𝑟 , which is another way of expressing 𝑛𝐶𝑟. The term number is always one more than 𝑟. e.g. Find the coefficient of the fourth term of 𝑥+𝑦 7 𝑟=4−1=3⇒ 7 3 =35

Compute Binomial Coefficients Find the coefficient of the third term of each of the following: 𝑥+𝑦 4 𝑥+𝑦 8 𝑥+𝑦 10 Find the coefficient of the fifth term of each of the following: 𝑎+𝑏 6 𝑚+𝑛 9

Compute Binomial Coefficients e.g. Find the coefficient of the third term of 2𝑥+𝑦 5 . 𝑟=3−1=2 5 2 =10 But! As one of the nomials is 2𝑥, the coefficient will include 2 5−2 = 2 3 =8 Overall coefficient is 10×8=80.