BackIn Exercises 49-53, use counting principles to find the probability.
51. A shipment of 200 calculators contains 3 defective units. What is the probability that a sample of three calculators will have
c. at least one defective calculator?
Selecting a Committee
Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly.
c. What is the probability that the committee is composed of three Democrats and four Republicans?
ATM You want to obtain cash by using an ATM, but it’s dark and you can’t see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first.
c. How many random selections are required to be absolutely sure that the card works because it is inserted correctly?
Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.
a. How many different ways can the five officers be appointed?
In Exercises 45-48, use combinations and permutations.
46. Five players on a basketball team must each choose one of the five players on the opposing team to defend. In how many ways can the players choose their defensive assignments?
50. Investment Committee A company has 200 employees, consisting of 144 women and 56 men. The company wants to select five employees to serve as an investment committee.
a. Use technology to find the number of ways that 5 employees can be selected from 200.
Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available.
a. If the subjects are selected and treated one at a time in sequence, how many different sequential arrangements are possible if 14 people are selected from the 16 that are available?
Cards In Exercises 59-62, you are dealt a hand of five cards from a standard deck of 52 playing cards.
62. Find the probability of being dealt three of a kind (the other two cards are different from each other).
6. A shipment of 250 netbooks contains 3 defective units. Determine how many ways a vending company can buy three of these units and receive
c. at least one good unit.
High Fives
b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes?
Combination Lock The typical combination lock uses three numbers, each between 0 and 49. Opening the lock requires entry of the three numbers in the correct order. Is the name “combination” lock appropriate? Why or why not?
Traveling Salesperson A salesperson must travel to eight cities to promote a new marketing campaign. How many different trips are possible if any route between cities is possible?
Selecting a Committee
Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly.
a. What is the probability that the committee is composed of all Democrats?"
50. Investment Committee A company has 200 employees, consisting of 144 women and 56 men. The company wants to select five employees to serve as an investment committee.
d. Explain how the company can select a representative sample of the male and female population of employees.
Text Twist
In the game Text Twist, six letters are given and the player must form words of different lengths using those letters. Suppose the letters in a particular game are ENHSIC.
a. How many different arrangements are possible using all 6 letters?