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Permutations - Arrangements Working out the number of elements in a sample space without having to list them IntroductionIntro ExerciseFactorials Factorial Practice

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Counting multiple events (including choices from different groups) The number of different possibilities from a multiple event situation can be determined by: (a) Listing all the possibilities (using common sense, a table or a tree diagram) (b) Using a box diagram.

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Counting Arrangements (Permutations) With The Box Method Notes: Instead of notes to copy into your book, we are going to work through 7 questions and then you can ask for extra notes if you feel that you need them. Remember to stick the example sheet into your workbook.

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Counting multiple events (including choices from different groups) Example 1 How many different possibilities are possible when tossing 3 coins HHHHHT HTHHTT THHTHT TTHTTT Method 1: Counting by listing Method 2: Counting using the box method 8 possibilities Coin 1Coin 2Coin 3 222 = 8

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Counting multiple events (including choices from different groups) Example 2 How many different possibilities are possible when rolling 3 dice. Method 1: Counting by listing Method 2: Counting using the box method D 1D 2D 3 666 = 216 Not plausible since too many alternatives

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Counting multiple events (including choices from different groups) Example 3 How many different possibilities are possible when choosing 3 courses off a menu with 3 different entrees, 7 main courses and 4 desserts. EMD 374 = 84

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Counting multiple events (including choices from different groups) Example 4 How many different standard number plates are possible in Victoria where we have 3 letters and 3 numbers. L1L2L3N1N2N3 26 10 = 17,576,000 L1L2L3N1N2N3 2526 10 = 16,900,000 Extra Question – How many standard number plates are available to the public?

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Counting Arrangements (Arrangement = Ordered group OR Permutation) Example 5 – Arranging Books (a) How many different arrangements (ordered groups or permutations) are possible of 6 books on a shelf B1B2B3B4B5B6 654321 = 720

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Counting Arrangements (Arrangement = Ordered group OR Permutation) Example 5 – Arranging Books (b) If 6 books are chosen from 10 titles and placed on a shelf, how many different arrangements (ordered groups or permutations) are possible on the shelf. B1B2B3B4B5B6 1098765 = 151,200

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Counting Arrangements (Arrangement = Ordered group OR Permutation) Example 6 – Arranging Letters How many ways can the letters in the word CAPTION be arranged: (a) With no conditions on letter placements? (b) Starting with a vowel? (c) Starting and ending with a consonant? L1L2L3L4L5L6L7 7654321 = 5040 VL1L2L3L4L5L6 3654321 = 2160 C1L1L2L3L4L5C2 4543213 = 1440

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Counting Arrangements (Arrangement = Ordered group OR Permutation) Example 6 – Arranging Letters How many ways can the letters in the word CAPTION be arranged: (d) With vowels and consonants in alternate positions? (e) Not starting with C? (f) Starting with the C and ending with N? C1V1C2V2C3V3C4 4332211 = 144 Not CL1L2L3L4L5L6 6654321 = 4320 CLL2L3L4L5N 1543211 = 120

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Counting Arrangements (Arrangement = Ordered group OR Permutation) Example 6 – Arranging Letters How many ways can the letters in the word CAPTION be arranged: (g) With consonants together? C1C2C3C4V1V2V3 4321321 = 144 V1C1C2C3C4V2V3 3432121 = 144 V1V2C1C2C3C4V3 3243211 = 144 V1V2V1C1C2C3C4 3214321 = 144 Total number of arrangements = 144 + 144 + 144+ 144 =576

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Counting Arrangements (Arrangement = Ordered group OR Permutation) Example 7 – Flag Problem A flag is made up of 4 coloured panels. If the possible colours are red, blue, white, green and yellow and no colour is used more than once how many arrangements are possible: (a) With no conditions? (b) With white at the start and any colour except green at the end? (c) With yellow as one of the stripes?

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Using the box method for working out the number of Permutations Ex 12B p354 Q3 - 15

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Factorials – A shortcut for multiplying boxes 3! = 3 x 2 x 1 = 6 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 5! = 120 10! = 3,628,800 7654321 = 7! = 5040

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Factorial Fractions 1211109876

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Factorial Problems Important Fact 0! = 1 Ex12B p354 Work out Q1 & 2 with a calculator (keyboard Math/Calc) Work out Q1& 2 without a calculator

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Factorial Problems Homework Ex12B p354 Confirm your calculations in questions 3 to 15 using the factorial function on your calculator.

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