An arrangement of objects in which order is not important. Combinations

Examples where order does not matters Choosing three students to help carry in a new supply of Phy. Ed equipment Choosing 4 toppings on your pizza Choosing 10 students for a special summer music camp

Combinations formula and notations Factorial Symbol is n! where n is any number 3! = 3•2•1 = 6 6! = 6•5•4•3•2•1 = 720 0! = 1 𝑛 𝐶 𝑟 = 𝑛! 𝑛−𝑟 ! 𝑟! ; where n is the number of objects taken r at a time. Example: in a class of 20 students, three students will represent the class on the Student Advisory Board. How many different sets could you have? 20 𝐶 3 = 20! 20−3 !•3! = 20•19•18•17•16•15•14•13•12•11•10•9•8•7•6•5•4•3•2•1 17•16•15•14•13•12•11•10•9•8•7•6•5•4•3•2•1 • 3•2•1 = 20•19•18 3•2•1 = 1140 ways

Your band has written 12 songs and plans to record 8 of them for a CD Your band has written 12 songs and plans to record 8 of them for a CD. In how many ways different sets of songs can you have on the CD if you don’t care what order they are in? Find the number of ways 8 songs can be picked for a CD taken from a total of 12 songs. 12 𝐶 8 = 12! 12−8 !•8! = 12! 4!•8! = 12•11•10•9•8•7•6•5•4•3•2•1 4•3•2•1 • 8•7•6•5•4•3•2•1 = 495 ways What is the probability of your and your best friend’s favorite songs will be the chosen? You don’t care which other songs are chosen: 𝑤ℎ𝑎𝑡 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 10 𝐶 6 12 𝐶 8 = 210 495 = 14 33 What is the probability that your favorite song will be first and your friend’s will be second on the CD? Need to find: 1 12 𝑃 2 12 𝑃 2 = 12! 12−2 ! = 12! 10! = 12•11 = 132 ways. So your chances of being first, friend second is = 1 132

Probability and Combinations 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑛 𝐶 𝑟 = 𝑛! 𝑛−𝑟 ! 𝑟! 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑛 𝐶 𝑟 = 𝑛! 𝑛−𝑟 ! 𝑟! Probability and Combinations You order a sandwich at a restaurant. You can choose 2 side dishes from a list of 8. How many combinations of side dishes are possible? 8 𝐶 2 = 8! 8−2 !•2! = 8! 6!•2! = 8•7•6•5•4•3•2•1 6•5•4•3•2•1 ∗ 2•1 = 28 different combinations of side dishes What are the probability you ordered the same two side dishes that your friend orders? There would be only one possible combination that is the same so the probability of ordering the same as your friend is 1 28 .

Probability and Combinations 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑛 𝐶 𝑟 = 𝑛! 𝑛−𝑟 ! •𝑟! Spring City Band Night Parade is coming soon. Two out of 10 flag twirlers are randomly chosen to lead the flag twirlers in the school parade. What is the probability that it will be you and your best friend? Find the sample space, how many ways can 2 people can be chosen? 10 𝐶 2 = 10! 10−2 !•2! = 10•9 2 = 45 ways There is only one combination that would make you and your friend the chosen leaders is 1 45

Worksheet 13.3