See Problem 5.

For each situation, determine whether to use a permutation or a combination. Then solve the problem.

39. How many different teams of 11 players can be chosen from a soccer team of 16?
40. Suppose you find seven equally useful articles related to the topic of your research paper. In how many ways can you choose five articles to read?
41. A salad bar offers eight choices of toppings for a salad. In how many ways can you choose four toppings?

B Apply

Assume a and b are positive integers. Determine whether each statement is true or false. If it is true, explain why. If it is false, give a counterexample.

42. a! + b! = b! + a!
43. a!(b!c!) = (a!b!)c!
44. (a + b)! = a! + b!
45. (ab)! = a!b!
46. ( a ! ) ! = ( a ! ) 2 open eh factorial close factorial equals . open eh factorial close squared
47. ( a ! ) b = a ( b ! ) open eh factorial close to the b . equals . eh super open b factorial close end super
48. Think About a Plan You and your friends are picking up videos at a video store. You have selected 7 videos but will only have time to watch 3 videos together. How many different ways can you select the 3 videos to watch?
    • Does the order in which the videos are selected make a difference?
    • What formula should you use?

49. Security A car door lock has a five-button keypad. Each button has two numerals. The entry code 21914 uses the same button sequence as the code 11023. How many different five-button patterns are possible? You can use a button more than once.

    

    1. 120
    2. 720
    3. 3125
    4. 5555
50. Consumer Issues A consumer magazine rates televisions by identifying two levels of price, five levels of repair frequency, three levels of features, and two levels of picture quality. How many different ratings are possible?
51. Writing In how many ways is it possible to arrange the two numbers a and b in an ordered pair? Explain why such a pair is called an ordered pair.
52. Reasoning Determine whether the statement   n C r =   n P r sub n , cap c sub r , equals , sub n , cap p sub r  is always, sometimes, or never true. Explain your reasoning.

53. There are 3!, or 6, arrangements of 3 objects. Consider the number of clockwise arrangements possible for objects placed in a loop, without a beginning or end.

    ABC, BCA, and CAB are all parts of one possible clockwise loop arrangement of the letters A, B, and C.

    1. Find the number of clockwise loop arrangements possible for letters A, B, and C.
    2. Use the diagram below to help find the number of loop arrangements possible for A, B, C, and D.
    3. Write an expression for the number of clockwise loop arrangements for n objects.

    

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