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Chapter 5 Section 4 The Multiplication Principle

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Generalized Multiplication Principle If a task consists of t operations performed consecutively, then the task can be performed in m 1 m 2 m 3 m 4 … m t ways. Trick is to identify the task and the operations.

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Exercise 3 (page 228) How many two-letter words (including non- sense words) can be formed? Note: 1.Task: Form two-letter word 2.Two operations 1.Write the first letter 2.Write the second letter 3.Restriction(s): None Solution:26 26 = 676

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Exercise 7 (page 228) Have 5 different pairs of gloves How many ways can you select a right-handed glove and a left-handed glove such that they don’t match? Note: 1.Task: Select two gloves 2.Two operations 1.Pick the right-handed glove 2.Pick the left-handed glove 3.The authors assume that you pick the right- handed glove first. 4.Restriction(s): The two gloves can not match

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Exercise 7 Solution __ __ = # of ways For selecting the right-handed glove: 5 _ For selecting the left-handed glove: 5 4 Answer: 5 4 = 20

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Exercise 11 (page 228) Toss a coin 6 times and observe the sequence of heads and tails. How many different sequences? Note: 1.Task: Observe the sequence of heads and tails 2.Six operations 1.First coin toss 2.Second coin toss (etc.) 6.Sixth coin toss 3.Restriction(s): None

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Exercise 11 Solution Task: __ __ __ __ __ __ Answer: 2 2 2 2 2 2 = 2 6 = 64

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Exercise 23 Part (b) (page 229) Set: { h, o, t, s, m, x, e } How many four letter words (including nonsense words) can be formed with the restriction: Letters cannot repeat. Solution: 7 6 5 4 = 840

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Exercise 23 Part (c) (page 229) Set: { h, o, t, s, m, x, e } How many four letter words (including nonsense words) can be formed with the restriction: Begins with ‘h’ and repetitions are allowed. Solution: 1 7 7 7 = 343

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Exercise 23 Part (d) (page 229) Set: { h, o, t, s, m, x, e } How many four letter words (including nonsense words) can be formed with the restrictions: Word must end in vowel and repetitions are not allowed. Solution: 6 5 4 2 = 240

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