SECTION 13.2 Permutations and Combinations 923 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 13.2 Assess Your Understanding 2. Multiple Choice The binomial coefficient 6 4 ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ equals (pp. 899–90 1) (a) 6! 4! (b) 6! 4! 2! ⋅ (c) 6! 2! (d) ( ) −6 4 ! 2! 1. = 0! ; = 1! (p. 858) In Problems 7–14, find the value of each permutation. 7. ( ) P 6, 2 8. ( ) P 7, 2 9. ( ) P 4, 4 10. ( ) P 8, 8 11. ( ) P 7, 0 12. ( ) P 9, 0 13. ( ) P 8, 4 14. ( ) P 8, 3 In Problems 15–22, use formula (2) to find the value of each combination. 15. ( ) C 8, 2 16. ( ) C 8, 6 17. ( ) C 7, 4 18. ( ) C 6, 2 19. ( ) C 15, 15 20. ( ) C 18, 1 21. ( ) C 26, 13 22. ( ) C 18, 9 Skill Building 23. List all the permutations of 5 objects a b c d , , , ,and e choosing 3 at a time without repetition. What is ( ) P 5, 3 ? 24. List all the permutations of 5 objects a b c d , , , ,and e choosing 2 at a time without repetition. What is ( ) P 5, 2 ? 25. List all the permutations of 4 objects 1, 2, 3, and 4 choosing 3 at a time without repetition. What is ( ) P 4, 3 ? 26. List all the permutations of 6 objects 1, 2, 3, 4, 5, and 6 choosing 3 at a time without repetition. What is ( ) P 6, 3 ? 27. List all the combinations of 5 objects a b c d , , , ,and e taken 3 at a time. What is ( ) C 5, 3 ? 28. List all the combinations of 5 objects a b c d , , , ,and e taken 2 at a time. What is ( ) C 5, 2 ? 29. List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is ( ) C 4, 3 ? 30. List all the combinations of 6 objects 1, 2, 3, 4, 5, and 6 taken 3 at a time. What is ( ) C 6, 3 ? 31. Forming Codes How many two-letter codes can be formed using the letters A B C , , ,and D? Repeated letters are allowed. 32. Forming Codes How many two-letter codes can be formed using the letters A B C D , , , ,and E? Repeated letters are allowed. 33. Forming Numbers How many three-digit numbers can be formed using the digits 0 and 1? Repeated digits are allowed. 34. Forming Numbers How many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9? Repeated digits are allowed. Applications and Extensions 35. Lining People Up In how many ways can 4 people be lined up? 36. Stacking Boxes In how many ways can 5 different boxes be stacked? 37. Forming Codes How many different three-letter codes are there if only the letters A B C D , , , ,and E can be used and no letter can be used more than once? 38. Forming Codes How many different four-letter codes are there if only the letters A B C D E ,,, ,,and F can be used and no letter can be used more than once? 39. Stocks on the NYSE Companies whose stocks are listed on the New York Stock Exchange (NYSE) have their company name represented by 1, 2, or 3 letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NYSE? 40. Stocks on the NASDAQ Companies whose stocks are listed on the NASDAQ stock exchange have their company name represented by either 4 or 5 letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NASDAQ? 41. Establishing Committees In how many ways can a committee of 4 students be formed from a pool of 7 students? 42. Establishing Committees In how many ways can a committee of 3 professors be formed from a department that has 8 professors? 43. Possible Answers on a True/False Test How many arrangements of answers are possible for a true/false test with 10 questions? Concepts and Vocabulary 3. A(n) is an ordered arrangement of r objects chosen from n objects. 4. A(n) is an arrangement of r objects chosen from n distinct objects, without repetition and without regard to order. 5. ( ) = P n r , 6. ( ) = C n r , 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
RkJQdWJsaXNoZXIy NjM5ODQ=