1. Notation What does the symbol ! represent? The five starting players of an NBA basketball
team can stand in a line 5! different ways, so what is the actual number of ways that the five
players can stand in a line?
2. Delaware Multi-Win Lotto In the Delaware Multi-Win Lotto game, a bettor selects six different numbers, each between 1 and 35. Winning the top prize requires that the selected num-bers match those that are drawn, but the order does not matter. Do calculations for winning this lottery involve permutations or combinations? Why?
3. Delaware Play 3 In the Delaware Play 3 lottery game, a bettor selects three numbers be-tween 0 and 9 and any selected number can be used more than once. Winning the top prize re-quires that the selected numbers match those and are drawn in the same order. Do calculations
for this lottery involve the combinations rule or either of the two permutation rules presented in
this section? Why or why not? If not, what rule does apply?
4. 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?
In Exercises 5–36, express all probabilities as fractions.
5. Pin Numbers Use of ATM cards and hotel safes typically requires a four-digit (each 0
through 9) code, such as 3312. Digits can be repeated, but they must be entered in the correct
order. If someone gains a code that was randomly selected, what is the prob-ability of getting the correct code on the first try?
6. Social Security Numbers A Social Security number consists of nine digits in a particular
order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if
you randomly select the other digits, what is the probability of getting the correct Social Secu-rity number of the person who was given the receipt?
7. Quinela In a horse race, a quinela bet is won if you selected the two horses that finish first
and second, and they can be selected in any order. The 144th running of the Kentucky Derby
had a field of 20 horses. What is the probability of winning a quinela bet if you randomly select
the horses?
8. Soccer Shootout In the FIFA Women’s World Cup 2019, a tie at the end of two overtime
periods leads to a “shootout” with five kicks taken by each team from the penalty mark. Each
kick must be taken by a different player. How many ways can 5 players be selected from the
11 eligible players? For the 5 selected players, how many ways can they be designated as first,
second, third, fourth, and fifth?
9. Statistics Counts How many different ways can the letters of “statistics” be arranged? If
the letters of “statistics” are arranged in a random order, what is the probability that the result
will be “statistics”?
10. Radio Station Call Letters Radio and Television station call letters must begin with either K (for stations west of the Mississippi River) or W (for stations east of the Mississippi
River) and must include either two or three additional letters. How many different possibilities are there?
11. Scheduling Routes A presidential candidate plans to begin her campaign by visiting the
capitals of 4 of the 50 states. If the four capitals are randomly selected without replacement,
what is the probability that the route is Sacramento, Juneau, Hartford, and Bismarck, in that
order?
12. Survey Cross Validation One way to identify survey subjects who don’t take the survey
seriously is to repeat a question with similar wording. If a survey with 10 questions includes
three questions that are the same except for minor differences in wording, how many different
ways can the 10 questions be arranged?
13. Safety with Numbers The author owns a safe in which he stores all of his great ideas for
the next edition of this book. The safe “combination” consists of four numbers, with each number
from 0 to 99. The safe is designed so that numbers can be repeated. If another author breaks in and
tries to steal these ideas, what is the probability that he or she will get the correct combination on
the first attempt? Assume that the numbers are randomly selected. Given the number of possibili-ties, does it seem feasible to try opening the safe by making random guesses for the combination?
14. Electricity When testing for current in a cable with five color-coded wires, the author used
a meter to test two wires at a time. How many different tests are required for every possible
pairing of two wires?
15. Jumble Many newspapers carry “Jumble,” a puzzle in which the reader must unscramble
letters to form words. The letters MHRHTY were included in newspapers on the day this exercise was written. How many ways can those letters be arranged? Identify the correct unscram-bling, then determine the probability of getting that result by randomly selecting one a selecting one arrangement of the given letters.
16. DNA Nucleotides DNA (deoxyribonucleic acid) is made of nucleotides. Each nucleo-tide can contain any one of these nitrogenous bases: A (adenine), G (guanine), C (cytosine), T
(thymine). If one of those four bases (A, G, C, T) must be selected three times to form a linear
triplet, how many different triplets are possible? All four bases can be selected for each of the
three components of the triplet.
17. Powerball As of this writing, the Powerball lottery is run in 44 states. Winning the jackpot
requires that you select the correct five different numbers between 1 and 69 and, in a separate
drawing, you must also select the correct single number between 1 and 26. Find the probability
of winning the jackpot.
18. Teed Off When four golfers are about to begin a game, they often toss a tee to randomly
select the order in which they tee off. What is the probability that they tee off in alphabetical
order by last name?
19. ZIP Code If you randomly select five digits, each between 0 and 9, with repetition allowed,
what is the probability you will get the author’s ZIP code?
23. 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?
b. How many different ways can a committee of five be appointed?
c. What is the probability of randomly selecting the committee members and getting the five
youngest of the qualified candidates?
Answers for section 4:4
1. The symbol ! is the factorial symbol, which represents
the product of decreasing whole numbers, as in
5! =5# 4# 3# 2# 1=120. Five NBA players can stand in line
120 different ways.
3. Because repetition is allowed, numbers are selected with replace-ment, so the combinations rule and the two permutation rules do
not apply. The multiplication counting rule can be used to show
that the number of possible outcomes is 10# 10# 10=1000.
5. 1>10,000 7. 1>190
9. 50,400; 1>50,400 11. 1>5,527,200
13. 1>100,000,000. No, there are far too many different possibilities.
15. 360; 1>360 (RHYTHM)
17. 1>292,201,338 19. 1>100,000
21. Area codes: 792. Phone numbers: 6,272,640,000. Yes. (With a
total population of about 400,000,000, there would be roughly 16
phone numbers for every adult and child.)
23. a. 360,360