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Combinatorics is a branch of mathematics that deals with counting. Counting is important: Chapter 3. Sets & Combinatorics Section 3.2 e.g. How much money I have? What’s the size of my computer memory? Counting How may elements (integers) are between i and j? How many combinations to arrange 3 numbers? …… 1 2 e.g. A child can choose one jellybean out of jellybeans(red, black), and one gummy bear out of three gummy bears(yellow, green, white). How many different sets of candy can the child have? 2 stages: Y: R, Y R G: R, G W: R, W choose jellybean n1 outcomes for 1st event, n2 outcomes for 2nd event. Then there are n1 * n2 outcomes for the sequence of the 2 events. e.g. choose gummy bear Y: B, Y G: B, G W: B, W B 3 4 10 toppings available in a pizza store, and the customer can choose any # of toppings, how many different pizza the store can make? Addition principle: If A and B are disjoint events with n1 and n2 possible outcomes, respectively, then the total # of outcomes for event A or B = n1 + n2. Toppings 1 choose it or w/o it OR : 2 2 * 2 3 * 2 The last part of your phone # contains 4 digits. How many four-digit numbers are there? (Select a password of 4 digits, how many possible passwords?) 9933, 1982, 2017, 1016, 0007, … 10 * 10 * 10 * 10 = 10,000 different numbers outcomes ∴6 outcomes e.g. Multiplication principle: … * … 10 * e.g. 2 = 210 =1,024 A customer wants to buy a vehicle from a dealer. The dealer has 23 autos and 14 trucks in stock. How many selections the customer have? 23 + 14 = 37 10 10 10 10 C10 0 + C1 + C 2 + C 3 + ...C10 = 1,024 (In Section 3.4 Permutations and Combinations) 5 6 1 e.g. decision tree: | A ∪ B | = |A| + |B| if A and B are disjoint sets. e.g.: use x, y and z to construct strings of length 3. How many strings do not have a z following y. In general, | A ∪ B | = |A| + |B| - | A ∩ B | | A – B | = |A| - | A ∩ B | x (In Section 3.3 Principle of Inclusion and Exclusion) e.g. x y How many four-digit number begin with a 4 or 5? 2 disjoint cases: numbers begin with 4 = 1*10*10*10=1000 x y 3 z x + y 2 x + y 3 z y z y x zx y + 3 x z x y + 2 + x z x y 3 z y + y 2 + x y z 3 = 21 numbers begin with 5 = 1000 The answer is 1000 + 1000 = 2000 7 8 2