Consider a group of ndistinct individuals or objects (“distinct” means that there is some characteristic that differentiates any particular individual or object from any other). How many ways are there to select a subset of size kfrom the group? For example, if a Little League team has 15 players on its roster, how many ways are there to select 9 players to form a starting lineup? Or if a university bookstore sells ten different laptop computers but has room to display only three of them, in how many ways can the three be chosen?
An answer to the general question just posed requires that we distinguish between two cases. In some situations, such as the baseball scenario, the order of selection is important. For example, Angela being the pitcher and Ben the catcher gives a different lineup from the one in which Angela is catcher and Ben is pitcher.
Often, though, order is not important and one is interested only in which individuals or objects are selected, as would be the case in the laptop display scenario.
n1n2n3n45(4)(3)(3)(2)5 72
(Di, Pj, Qk) N5n1n2n35 (5)(12)(9)5540
k53 n1n2
#
c#
nk
Example 2.20
(Example 2.18 continued)
DEFINITION An ordered subset is called a permutation.The number of permutations of size kthat can be formed from the nindividuals or objects in a group will be denoted by Pk,n. An unordered subset is called a combination.One way to denote the number of combinations is Ck,n, but we shall instead use notation that is quite common in probability books: AnkB, read “nchoose k”.
The number of permutations can be determined by using our earlier counting rule for k-tuples. Suppose, for example, that a college of engineering has seven departments, which we denote by a, b, c, d, e, f,and g. Each department has one rep- resentative on the college’s student council. From these seven representatives, one is
Example 2.21
PROPOSITION Pk, n5 n!
(n2k)!
to be chosen chair, another is to be selected vice-chair, and a third will be secretary.
How many ways are there to select the three officers? That is, how many permuta- tions of size 3 can be formed from the 7 representatives? To answer this question, think of forming a triple (3-tuple) in which the first element is the chair, the second is the vice-chair, and the third is the secretary. One such triple is (a, g, b), another is (b, g, a), and yet another is (d, f, b). Now the chair can be selected in any of ways. For each way of selecting the chair, there are ways to select the vice- chair, and hence (chair, vice-chair) pairs. Finally, for each way of selecting a chair and vice-chair, there are ways of choosing the secretary.
This gives
as the number of permutations of size 3 that can be formed from 7 distinct individ- uals. A tree diagram representation would show three generations of branches.
The expression for P3,7can be rewritten with the aid of factorial notation.
Recall that 7! (read “7 factorial”) is compact notation for the descending prod- uct of integers (7)(6)(5)(4)(3)(2)(1). More generally, for any positive integer m, . This gives , and we also define . Then
More generally,
Multiplying and dividing this by gives a compact expression for the number of permutations.
(n2 k)!
Pk, n5n(n2 1)(n22)
#
c#
(n2(k22))(n2 (k2 1)) P3, 75 (7)(6)(5)5 (7)(6)(5)(4!)
(4!) 57!
4!
0! 51
1! 51 m!5 m(m2 1)(m22)
#
c#
(2)(1)P3, 75 (7)(6)(5)5 210 n35 5 73 65 42
n25 6
n15 7
There are ten teaching assistants available for grading papers in a calculus course at a large university. The first exam consists of four questions, and the professor wishes to select a different assistant to grade each question (only one assistant per question). In how many ways can the assistants be chosen for grading? Here
and . The number of permutations is
That is, the professor could give 5040 different four-question exams without using the same assignment of graders to questions, by which time all the teaching assis- tants would hopefully have finished their degree programs! ■ Now let’s move on to combinations (i.e., unordered subsets). Again refer to the student council scenario, and suppose that three of the seven representatives are to be selected to attend a statewide convention. The order of selection is not important;
all that matters is which three get selected. So we are looking for , the number of combinations of size 3 that can be formed from the 7 individuals. Consider for a
A73B P4,105 10!
(102 4)!5 10!
6! 510(9)(8)(7)5 5040 k5 subset size5 4
n5 group size510
moment the combination a,c,g. These three individuals can be ordered in ways to produce permutations:
Similarly, there are ways to order the combination b,c,eto produce permuta- tions, and in fact 3! ways to order any particular combination of size 3 to produce permutations. This implies the following relationship between the number of com- binations and the number of permutations:
It would not be too difficult to list the 35 combinations, but there is no need to do so if we are interested only in how many there are. Notice that the number of permuta- tions 210 far exceeds the number of combinations; the former is larger than the latter by a factor of 3! since that is how many ways each combination can be ordered.
Generalizing the foregoing line of reasoning gives a simple relationship between the number of permutations and the number of combinations that yields a concise expression for the latter quantity.
P3, 75(3!)?Q7 3R1Q7
3R5 P3, 7
3! 5 7!
(3!)(4!)5(7)(6)(5) (3)(2)(1) 535 3!5 6
a, c, g a, g, c c, a, g c, g, a g, a, c g, c, a
3! 56
PROPOSITION Qn
kR5 Pk,n
k! 5 n!
k!(n2 k)!
Notice that and since there is only one way to choose a set of (all) nelements or of no elements, and since there are nsubsets of size 1.
A particular iPod playlist contains 100 songs, 10 of which are by the Beatles.
Suppose the shuffle feature is used to play the songs in random order (the random- ness of the shuffling process is investigated in “Does Your iPod Really Play Favorites?” (The Amer. Statistician,2009: 263–268). What is the probability that the first Beatles song heard is the fifth song played?
In order for this event to occur, it must be the case that the first four songs played are not Beatles’ songs (NBs) and that the fifth song is by the Beatles (B). The number of ways to select the first five songs is 100(99)(98)(97)(96). The number of ways to select these five songs so that the first four are NBs and the next is a B is 90(89)(88)(87)(10). The random shuffle assumption implies that any particular set of 5 songs from amongst the 100 has the same chance of being selected as the first five played as does any other set of five songs; each outcome is equally likely.
Therefore the desired probability is the ratio of the number of outcomes for which the event of interest occurs to the number of possible outcomes:
Here is an alternative line of reasoning involving combinations. Rather than focus- ing on selecting just the first five songs, think of playing all 100 songs in random order. The number of ways of choosing 10 of these songs to be the Bs (without regard to the order in which they are then played) is . Now if we choose 9 of the last 95 songs to be Bs, which can be done in ways, that leaves four NBs and one B for the first five songs. There is only one further way for these five to start with
A959B A10010B P(1st B is the 5th song played)5 90
#
89
#
88
#
87
#
10 100
#
99
#
98
#
97
#
965 P4, 90
#
(10)P5, 100 5 .0679 An1B 5n
An0B 5 1 AnnB 51
Example 2.22
Example 2.23
four NBs and then follow with a B (remember that we are considering unordered subsets). Thus
It is easily verified that this latter expression is in fact identical to the first expres- sion for the desired probability, so the numerical result is again .0679.
The probability that one of the first five songs played is a Beatles’ song is
It is thus rather likely that a Beatles’ song will be one of the first five songs played.
Such a “coincidence” is not as surprising as might first appear to be the case. ■ A university warehouse has received a shipment of 25 printers, of which 10 are laser printers and 15 are inkjet models. If 6 of these 25 are selected at random to be checked by a particular technician, what is the probability that exactly 3 of those selected are laser printers (so that the other 3 are inkjets)?
Let . Assuming that any
particular set of 6 printers is as likely to be chosen as is any other set of 6, we have equally likely outcomes, so ,where N is the number of ways of choosing 6 printers from the 25 and is the number of ways of choosing 3 laser printers and 3 inkjet models. Thus . To obtain , think of first choosing 3 of the 15 inkjet models and then 3 of the laser printers. There are ways of choosing the 3 inkjet models, and there are ways of choosing the 3 laser printers;
is now the product of these two numbers (visualize a tree diagram—we are really using a product rule argument here), so
Let and define D5and
D6in an analogous manner. Then the probability that at least 3 inkjet printers are selected is
■ 5
a15 3b a10
3b a25
6b 1
a15 4b a10
2b a25
6b 1
a15 5b a10
1b a25
6b 1
a15 6b a10
0b a25
6b
5.8530 P(D3´D4´D5´D6)5P(D3)1 P(D4)1 P(D5)1P(D6)
D45 5exactly 4 of the 6 printers selected are inkjet models6 P(D3)5N(D3)
N 5
a15 3b a10
3b a25
6b 5
15!
3!12!
#
10!3!7!
25!
6!19!
5 .3083 N(D3)
A103B N(D3) A153B NN(D5 A325)6B
P(D3)5 N(D3)/N
D35 5exactly 3 of the 6 selected are inkjet printers6 5
a99 9b a100
10b 1
a98 9b a100
10 b 1
a97 9b a100
10b 1
a96 9b a100
10b 1
a95 9b a100
10b
5.4162 P(1st B is the 1st or 2nd or 3rd or 4th or 5th song played)
P(1st B is the 5th song played)5 a95
9b a100
10b
EXERCISES Section 2.3 (29–44)
29. As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com).
a. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits as well as letters are per- mitted as characters? [Note:A character length of three or more is now mandated.]
b. How many domain names are there consisting of three letters in sequence? How many of this length are there if either letters or digits are permitted? [Note:All are cur- rently taken.]
c. Answer the questions posed in (b) for four-character sequences.
d.As of April 2006, 97,786 of the four-character sequences using either letters or digits had not yet been claimed. If a four-character name is randomly selected, what is the probability that it is already owned?
30. A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries.
a. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?
b. If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this?
c. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?
d.If 6 bottles are randomly selected, what is the probabil- ity that this results in two bottles of each variety being chosen?
e. If 6 bottles are randomly selected, what is the probability that all of them are the same variety?
31. a. Beethoven wrote 9 symphonies, and Mozart wrote 27 piano concertos. If a university radio station announcer wishes to play first a Beethoven symphony and then a Mozart concerto, in how many ways can this be done?
b. The station manager decides that on each successive night (7 days per week), a Beethoven symphony will be played, followed by a Mozart piano concerto, followed by a Schubert string quartet (of which there are 15). For roughly how many years could this policy be continued before exactly the same program would have to be repeated?
32. A stereo store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component:
Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood Compact disc player: Onkyo, Pioneer, Sony, Technics Speakers: Boston, Infinity, Polk
Turntable: Onkyo, Sony, Teac, Technics
A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions:
a. In how many ways can one component of each type be selected?
b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?
c. In how many ways can components be selected if none is to be Sony?
d. In how many ways can a selection be made if at least one Sony component is to be included?
e. If someone flips switches on the selection in a com- pletely random fashion, what is the probability that the system selected contains at least one Sony component?
Exactly one Sony component?
33. Again consider a Little League team that has 15 players on its roster.
a. How many ways are there to select 9 players for the starting lineup?
b. How many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters?
c. Suppose 5 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players?
34. Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects.
a. How many ways are there to randomly select 5 of these key- boards for a thorough inspection (without regard to order)?
b. In how many ways can a sample of 5 keyboards be selected so that exactly two have an electrical defect?
c. If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?
35. A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the grave- yard shift. A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group (drawing 6 slips without replacement from among 45).
a. How many selections result in all 6 workers coming from the day shift? What is the probability that all 6 selected workers will be from the day shift?
b. What is the probability that all 6 selected workers will be from the same shift?
c. What is the probability that at least two different shifts will be represented among the selected workers?
d. What is the probability that at least one of the shifts will be unrepresented in the sample of workers?
36. An academic department with five faculty members nar- rowed its choice for department head to either candidate A or candidate B.Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for Aand two for B.If the slips are selected for tally- ing in random order, what is the probability that Aremains ahead of Bthroughout the vote count (e.g., this event occurs if the selected ordering is AABAB,but not for ABBAA)?
37. An experimenter is studying the effects of temperature, pres- sure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pres- sures, and five different catalysts are under consideration.
a. If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible?
b. How many experimental runs are there that involve use of the lowest temperature and two lowest pressures?
c. Suppose that five different experimental runs are to be made on the first day of experimentation. If the five are randomly selected from among all the possibilities, so that any group of five has the same probability of selec- tion, what is the probability that a different catalyst is used on each run?
38. A box in a certain supply room contains four 40-W light- bulbs, five 60-W bulbs, and six 75-W bulbs. Suppose that three bulbs are randomly selected.
a. What is the probability that exactly two of the selected bulbs are rated 75-W?
b. What is the probability that all three of the selected bulbs have the same rating?
c. What is the probability that one bulb of each type is selected?
d. Suppose now that bulbs are to be selected one by one until a 75-W bulb is found. What is the probability that it is necessary to examine at least six bulbs?
39. Fifteen telephones have just been received at an authorized service center. Five of these telephones are cellular, five are cordless, and the other five are corded phones. Suppose that these components are randomly allocated the numbers 1, 2, . . . , 15 to establish the order in which they will be serviced.
a. What is the probability that all the cordless phones are among the first ten to be serviced?
b. What is the probability that after servicing ten of these phones, phones of only two of the three types remain to be serviced?
c. What is the probability that two phones of each type are among the first six serviced?
40. Three molecules of type A,three of type B,three of type C, and three of type Dare to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC.
a. How many such chain molecules are there? [Hint:If the three A’s were distinguishable from one another—A1, A2, A3—and the B’s, C’s, and D’s were also, how many
molecules would there be? How is this number reduced when the subscripts are removed from the A’s?]
b. Suppose a chain molecule of the type described is ran- domly selected. What is the probability that all three molecules of each type end up next to one another (such as in BBBAAADDDCCC)?
41. An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.
a. How many different possible PINs are there if there are no restrictions on the choice of digits?
b. According to a representative at the author’s local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: (i) all four digits identical (ii) sequences of consecutive ascending or descending digits, such as 6543 (iii) any sequence start- ing with 19 (birth years are too easy to guess). So if one of the PINs in (a) is randomly selected, what is the prob- ability that it will be a legitimate PIN (that is, not be one of the prohibited sequences)?
c. Someone has stolen an ATM card and knows that the first and last digits of the PIN are 8 and 1, respectively. He has three tries before the card is retained by the ATM (but does not realize that). So he randomly selects the 2ndand 3rddigits for the first try, then randomly selects a differ- ent pair of digits for the second try, and yet another ran- domly selected pair of digits for the third try (the individual knows about the restrictions described in (b) so selects only from the legitimate possibilities). What is the probability that the individual gains access to the account?
d.Recalculate the probability in (c) if the first and last dig- its are 1 and 1, respectively.
42. A starting lineup in basketball consists of two guards, two forwards, and a center.
a. A certain college team has on its roster three centers, four guards, four forwards, and one individual (X) who can play either guard or forward. How many different starting lineups can be created? [Hint: Consider lineups without X, then lineups with X as guard, then lineups with X as forward.]
b. Now suppose the roster has 5 guards, 5 forwards, 3 cen- ters, and 2 “swing players” (X and Y) who can play either guard or forward. If 5 of the 15 players are ran- domly selected, what is the probability that they consti- tute a legitimate starting lineup?
43. In five-card poker, a straight consists of five cards with adja- cent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card 10? What is the probability that it will be a straight?
What is the probability that it will be a straight flush (all cards in the same suit)?
44. Show that . Give an interpretation involving subsets.
AnkB5An2n kB