Chris Morgan, MATH G160 January 25, 2012 Lecture 7 Chapter 4.1: Combinatorics and Basic Counting Rules 1

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Combinatorics Many events have far too many outcomes to list all of them: - How many possible outcomes are there in flipping a coin six times? (64) - How many possible ways can I get a Jimmy Johns sandwich made? (a lot) - How many ways can I rearrange all of you in your desks? (even more ways) 3

Combinatorics Listing all possible situations is unpractical, and usually we dont care about each individual outcome (only care about total outcomes) Combinatorics is the study of counting rules so we can count more quickly 4

Example (I) Cassie owns 2 different pairs of shoes, 4 different shirts, and 3 different pairs of pants. How many different outfits can she wear? 2 * 4 * 3 = 24 shoes shirts pants 5

Example (I) ShoesShirtsPants ShirtsPants Shirts 6

Example (I) ShoesShirtsPants ShirtsPants Shirts 7

Basic Counting Rule Suppose that r actions (choices, experiments) are to be performed in a definite order, further suppose that there are m1 possibilities for the first action, m2 possibilities for the second action, etc, then there are m1 × m2 ×... × mr total possibilities altogether. 8

Basic Counting Rule To start, lets demonstrate BCR by tossing a coin twice: How many possible ways are there to toss the coin? - m1 = 2 (Heads or Tails) - m2 = 2 (Heads or Tails) 2 * 2 = 4 HT HT 9

Basic Counting Rule A phone number consists of 10 numbers. The prefix for Indianapolis is 317, how many phone numbers are possible? 10 digits, but the first three are set: _ _ _ _ _ _ _ 1 * 1 * 1 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 10 7 = 10,000,000 10

Basic Counting Rule What is the probability that a phone number contains the numbers 0-6 in any order? _ _ _ _ _ _ _ = ?? 7* 6 * 5 * 4 * 3 * 2 * 1 = 5,040 P(0-6) = 5, 040 / 10,000,000 =

Assuming Mary has 6 pairs of shoes, 10 different tops, 8 different bottoms and 4 different outwears, then how many combinations can she have for outfit? Mary is having a job interview and she wants to decide what to wear. If there are 2 pairs of shoes, 3 tops, 2 bottoms and 2 outwears that are appropriate for an interview and she randomly picks what to wear for the interview among all she has, what is the probability that she wears an interview-appropriate outfit? Example (II) 12

Mary bought a lock for her new bike since her last one was stolen. There are 4 slots numbered 0 to 9, how many possible combinations are there? If the combination only includes even numbers If the first number can not be 0 and all four numbers must all be different If the combination must have at least one 4 or at least one 5 What are the probabilities for those specified combination? Example (III) 13

At a Subway, you have to decide what you want to put in your sandwich, the choices you have are: Four types of bread Five types of cheese Six types of veggies Seven types of meat Assuming you can only choose one from each of the above categories, how many total possible combinations could we get? If I dont like white bread, only like Swiss cheese, dont like onion, and am allergic to seafood and chicken, what is the chance that I get a sandwich that I actually like? Example (IV) 14

An assiduous student named Sam finds himself hungry at 2 a.m. on a Tuesday. This unremitting undergraduate has become conscious of a considerable craving for Connies pizza. Alas, he is nowhere near Chicago to fulfill such a phenomenal food fantasy. He has to settle for 1 of 5 pizza places that still permits pie purchases. Each restaurant has 3 choices for crust type: thin crust, regular, and deep dish. Additionally, a customer is allowed to have at most 1 meat out of the 4 total choices and at most 1 vegetable out of the 5 for his toppings. How many possible pizzas could the famished freshman feast on? Ceteris paribus, how many possible pizzas are there if the place permits at most 2 meat choices? Ceteris paribus, how many pizzas are possible if the restaurant allows at most 2 meat choices and any combination of vegetables? Example (V) 15

A young man wants to plan a nice date for his girlfriend. He has the option of going to Chicago, Indianapolis, or staying Lafayette. If he chooses Chicago, he has 10 choices for a play and 100 choices for a restaurant. If he opts of Indianapolis, he has 5 choices for a play and 50 choices for a restaurant. If he remains in Lafayette, he only has 2 options for a play 20 choices for a restaurant. How many options does this gentleman have for a romantic evening out? What about if we added the option of Fort Wayne and this particular city boasted 8 plays and 75 restaurants? Example (VI) 16

N is population size (sampling) and n is sample size Sampling with replacement: How many possible ordered samples of size 3 with replacement from a population of size 4? In general, how many possible ordered samples of size n with replacement from a population of size N. Example (VII) 17