Section 2 Tree Diagrams Questions about homework? Submit homework! 3/24/2016 MATH 106, Section 2 1 Using “brute force” to solve problems • Some simple problems can be solved using “brute force” • List all possible arrangements and count them • The hard part – making sure that the list includes each one exactly once • Don’t miss any • No duplicates • You are going to flip a coin three times. How many possible outcomes are there? Let’s list them. HHH HHT HTH THH HTT THT TTH TTT • Let’s use a tree diagram (a pictorial representation) to get all of the possibilities. 3/24/2016 MATH 106, Section 2 2 What we mean by a “tree” … 1st toss 2nd toss 3rd toss heads heads • A common point called the root • The lines are the branches • The more parts there are, the taller the tree is • We will grow our trees to the right! tails heads heads tails tails heads heads tails tails heads tails Let’s do more examples … 3/24/2016 MATH 106, Section 2 tails 3 #1 You have 4 shirts (red, white, blue, and khaki) and 3 pairs of pants (khaki, black, and blue jeans). Construct a tree diagram to find how many different outfits can you make. Does it matter whether you pick shirts first or pants first to build the tree diagram? shirts pants red white blue khaki 3/24/2016 khaki black blue jeans khaki black blue jeans khaki black blue jeans khaki black blue jeans The number of different outfits is 12. The answer is (of course) the same if we pick pants first to build the tree diagram. MATH 106, Section 2 4 How many different outfits can you make, if you never have the same color shirt and pants? The number of different outfits without the same color shirt and pants is 12 – 2 = 10. 3/24/2016 MATH 106, Section 2 5 #2 A hot dog cart has the following condiments: ketchup, mustard, and relish. How many different “kinds” of hot dogs can you make? ketchup mustard relish yes yes no yes yes no no The number of different “kinds” of hot dogs can you make is 8. yes yes no yes no no no 3/24/2016 MATH 106, Section 2 6 How many different “kinds” of hot dogs can you make with exactly 2 toppings? The number of different “kinds” of hot dogs can you make with exactly 2 toppings is 3. 3/24/2016 MATH 106, Section 2 7 Homework Hints: In Section 2 Homework Problem #8, don’t be confused or fooled by the way the sentence is worded. Consider “cream” possibilities and “sugar” possibilities separately. In Section 2 Homework Problem #9, consider each topping separately. In Section 2 Homework Problems #11 and #12, consider possibilities for die#1 followed by possibilities for die#2 in a tree diagram. 3/24/2016 MATH 106, Section 2 8