11.7 The Fundamental Counting Principle and Permutations 729 7. The notation used to express the number of permutations of five items taken three at a time is ___________. P5 3 8. The notation used to express the number of permutations of seven items taken two at a time is ___________. P7 2 Practice the Skills In Exercises 9–20, evaluate the expression. 9. 6! 720 10. 8! 40,320 11. 0! 1 12. 1! 1 The correct sequence of five buttons must be pressed to unlock the door. a) How many different sequences of five buttons are possible (repetition is permitted)? 5 3125 5 = b) If a sequence of five buttons is randomly pressed, determine the probability that the sequence unlocks the door. 1 3125 0.00032 = 29. License Plate Numbers In Indiana, a standard car license plate number consists of 3 digits followed by 3 uppercase letters. How many different license plate numbers are possible if a) repetition of characters is allowed? 17,576,000 b) repetition of characters is not allowed? 11,232,000 30. License Plate Numbers In Wisconsin, a standard license plate number consists of 3 letters followed by 4 digits. How many different license plate numbers are possible if a) repetition of characters is allowed? 175,760,000 b) repetition of characters is not allowed? 78,624,000 31. Geometric Shapes Consider the five figures shown. In how many different ways can the figures be arranged a) from left to right? 120 b) from top to bottom if placed one under the other? 120 c) from left to right if the triangle is to be placed on the far right? 24 d) from left to right if the circle is to be placed on the far left and the triangle is to be placed on the far right? 6 32. Arranging Pictures The six pictures shown are to be placed side by side along a wall. 13. P3 2 6 14. P4 3 24 15. P8 0 1 16. P6 0 1 17. P4 4 24 18. P3 3 6 19. P8 4 1680 20. P9 5 15,120 Problem Solving 21. ATM Codes To use an automated teller machine, you generally must enter a four-digit code, using the digits 0–9. How many four-digit codes are possible if repetition of digits is permitted? 10,000 22. Social Security Numbers A social security number consists of nine digits. How many different social security numbers are possible if repetition of digits is permitted? 1,000,000,000 23. Outfits Juan has three ties, four shirts, and two pairs of pants. How many different outfits can he wear if he chooses one tie, one shirt, and one pair of pants for each outfit? 24 24. Coffee Choices When pouring a cup of coffee at work, Amelia has a choice of regular or decaffeinated. She can then choose to add half and half, soy milk, almond milk or no creamer at all. She then can choose to add sugar, stevia, monk fruit extract, honey, agave nectar, or no sweetener at all. How many different ways can Amelia prepare her coffee? 48 25. Computer Systems At a computer store, Imelda is considering 5 different computers, 4 different monitors, 7 different printers, and 2 different soundbars. Assuming that each of the components is compatible with one another and that one of each is to be selected, determine the number of different computer systems possible. 280 26. Selecting Furniture The Johnsons just moved into their new home and are selecting furniture for the family room. They are considering 5 different sofas, 2 different chairs, and 6 different tables. They plan to select one item from each category. Determine the number of different ways they can select the furniture. 60 27. Smartphone Passcode A passcode on a smartphone consists of 6 digits, and repetition of digits is allowed. a) Determine the number of possible six-digit passcodes. 1,000,000 b) If a person finds a smartphone and randomly enters 6 digits, what is the probability that the correct passcode is entered? 1 1,000,000 0.000001 = 28. Car Door Locks Some doors on cars can be opened by pressing a correct sequence of buttons. A display of the five buttons by the door handle of a car follows. 1 - 2 3 - 4 5 - 6 7 - 8 9 - 0 Copyright © 2022 by Jef Poskanzer <jef@ mail.acme.com>. All rights reserved.
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