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Counting Basics: Be careful of the boundary conditions Try to come up with a general rule

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Example #1 How many integers are less than 600 but greater than 500? Be careful of the boundary, not to include 500 nor 600. Answer: 599 – 500 = 99

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Example #2 Jim has a yard that is 60 yard long and 30 yard wide. If he places a host at each corner and places other posts three yards apart along the edge. How many posts are placed surrounding the yard? Be careful not to over count corner posts Set the pattern: each line with one corner

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1*3, 2*3, …10x3; total 10 posts 10 posts 1*3, 2*3, …, 20*3; total 20 posts 20 posts Answer: 20 + 10 + 20 + 10 = 60 posts

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Venn Diagrams Useful when counting among categories that may have overlapping entries Working from inside-out helps

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Example #3 At a party gathering, 19 person have a brother, 15 person have a sister, 7 person have both a brother and a sister, and 6 person don't have any siblings at all. How many person are at the party? Recognizing there are 2 categories: having a brother, having a sister. Draw the Venn diagrams accordingly.

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Answer: 7 + 12 + 8 + 6 = 33 Only have a brother: 19 – 7 = 12 Only have a sister: 15 – 7 = 8 Have no brother nor sister: 6 Have both brother & sister: 7

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Example #4 Every student in a classroom has at least one pet. 30 have a cat, 28 have a dog, and 26 have fish. If 13 students have fist and a cat, 15 students have fish and a dog, 11 students have both a cat and a dog, and 4 students have a cat, a dog, and fish. How many students in the classroom? Recognizing there are 3 categories: have a cat, have a dog, or have fish. Draw the Venn diagrams accordingly.

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Answer: 4 + 7 + 9+ 11 + 8+ 8 + 2 = 49 4 11 – 4 = 7 13 – 4 = 9 15 – 4 = 11 Dog only: 30–7–4–11 = 8 Cat only: 28-7-4-9 = 8 Fish only: 26 –4–9–11 = 2

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Example #5 How many of the smallest 1000 positive integers are divisible by 5, 6, or 7? Recognizing there are 3 categories: div by 5, div by 6, div by 7. Draw the Venn diagrams accordingly.

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Answer: (div-by-5: 200) + (div-by-6: 166) + (div-by-7: 142) - (div-by-5&6: 33) - (div-by-5&7: 28) - (div-by-6&7: 23) + (div-by-5&6&7: 4) 4 Div by 5&6: 33 div by 6&7: 23 div by 5&7: 28 Div by 5 only Div by 6 only Div by 7 only = 200 + 166 + 142 – 33 – 28 – 23 + 4 = 428

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Bowling pins and handshakes How many pins in the diagram? Answer: 1 + 2 + 3 + 4 = 10 What about: 1 + 2 + 3 + … + 100 = ?

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Example #6: 1 + 2 + 3 + … + 100 = ? Assume S = 1 + 2 + 3 + … + 100 we can also write S = 100 + 99 + 98 + … + 1 pair them as shown in circles, we got: 2 * S = 101 * 100 Thus S = 101 * 1000 / 2 = 5050

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Fundamental counting principle If there are m ways that one event can happen, and n ways a second event can happen, then there are m*n ways that both events can happen.

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Example #7 How many 10 digit whole numbers use only the digits 1 and 0? Note that the first digit can’t be 0, and all other digits have 2 choices. Answer: 1 * 2 9 = 512

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Example #8 How many squares can be formed by 4 of the dots in the unit grid as vertices? Keeping organized. Recognize all possible category

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Example #8 # of unit square: 9 # of 2x2 square: 4 # of 3x3 square: 1 # of 2x2 diagonal square: 2 Answer: 9 + 4 + 1 + 2 + 4 = 20 # of 1x1 diagonal square: 4

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Factorials and permutations n ! = n * (n – 1) * (n – 2 ) … * 2 * 1 Example: How many different "words" can be formed by re-arranging the letters in the word "COUNT"? Answer: 5! = 5 * 4 * 3 * 2 * 1 = 120

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Example # 9 How many different 8 letter "words" can be formed by re-arranging the letters in the word "GEEEETRY"? Think of E 1 and E 2 as two different characters first, we got total # of words: 8! Then remove the duplicates, we get the answer: 8! / 2 = 20160

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Permutation with restrictions How many even five digit numbers contains each of the digits 1 through 5? Working from the last digit, we got: answer = 2 * 4 * 3 * 2 * 1 = 48

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Combinations Permutation where the sequence of the elements doesn’t count. Can be calculated by removing the repeated ones from the permutation result. Combination = (permutation of the whole set) / (permutation of the selected set)

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Example #10 Remy wants to drink 3 different sodas from a list of 8 sodas. How many different soda combinations he can choose to drink? Permutation of the whole set: 8 * 7 * 6 Permutation of selected set: 3 * 2 * 1 # of combinations: (8 * 7 * 6) / (3 * 2 * 1) Answer: 56

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Example #11 Tracing the lines starting from A on the unit grid below, how many distinct 7-unit paths are there from A to B?

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Example #11 Must move 3 Up-moves in seven moves, and 4 Right moves in seven moves. Thus the question becomes: How many ways to put 4 Rs in 7 slots. Ways to put 4 Rs in 7 slots: 7 * 6 * 5 * 4 ; divide by the repeated ones: 4 * 3 * 2 * 1, we got the answer: 7*6*5*4 / (4*3*2*1) = 35

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Complementary counting Total # of desired = total # - the # that we don’t want (Hint: use this strategy if the desired set is a union of different sets)

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Example #11 Paul flits a fair coin eight times. In how many ways can he flip at least two heads? Total count: 2 8 ; Count for no head: 1 Count for 1 head: 8 Use complementary counting, the answer is: 2 8 – 1 – 8 = 247

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