week 7 - ics 6d
Dave will swim one day, run one day, and bike another day in a week. He does at most one activity on any particular day. How many ways are there for him to select his workout schedule (i.e. which activities he does which days)?
P(7,3) First Dave picks his swimming day, then his running day, and then his biking day. Since each activity is different, the order in which he selects the days matters. Therefore the number of selections is P(7, 3).
How to find the number of r permutations in a set of size n?
P(n, r)
One possible outcome of the experiment of dealing a 5-card hand would be??
(2H, 2D, 8S, JC, AC)
Consider a function that maps 5-permutations from a set S = {1, 2, ..., 20} to 5-subsets from S. The function takes a 5-permutation and removes the ordering on the elements. How many 5-permutations map on to the subset {2, 5, 13, 14, 19}? What is k? (3 steps)
120 or 5! 1) How many five permutations are there from set S? P(20, 5) = 27907200 2) How many five subsets are there? P(20,5)/5! or 20!/5!(15!) = 232560 3) 27907200/232560 = 120 alternatively, There are 5! ways to order five elements in a subset. Therefore, the function mapping 5-permutations to 5-subsets is k-to-1 for k = 5! = 120.
how many ranks with one suit?
13
Consider two closely related counting problems: 1.) A family goes to the animal shelter to adopt 3 cats. The shelter has 20 different cats from which to select. How many ways are there for the family to make their selection? 2.) Three different families go to the animal shelter to adopt a cat. Each family will select one cat. How many ways are there for the families to make their selections? Note that which family gets which cat matters.
1.) C(20, 3) 2.) P(20, 3)
In an experiment consisting of three flips of a fair coin, what is the probability that the first two flips are the same?
1/2 The event is the set E = {HHH, HHT, TTH, TTT}. |E|/|S| = 4/8 = 1/2.
In an experiment consisting of three flips of a fair coin, what is the probability that the first two flips are both heads?
1/4 The event is the set E = {HHH, HHT}. |E|/|S| = 2/8 = 1/4. Alternatively, since the H's are chosen then that leaves two options for the other character and we know the total is 8 so...
Ten members of a wedding party are lining up in a row for a photograph. (a) How many ways are there for the wedding party to line up in which the bride is not next to the groom?
10! - (2 * 9!) a1) total ways to line up is 10! a2) ways to line up in which bride IS next to the groom are 2 * 9! a3) 10! - (2 * 9!)
Ten members of a wedding party are lining up in a row for a photograph. (c) How many ways are there for the wedding party to line up if the groom is not at one end (i.e. in the leftmost or rightmost positions)?
10! - (9!+9!) c1) total ways 10! c2) ways in which groom is in leftmost 9! + 9!
Ten members of a wedding party are lining up in a row for a photograph. (b) How many ways are there for the wedding party to line up if the groom is not in the leftmost position?
10!-9! b1) total ways to line up is 10! b2) ways to line up in which groom is in leftmost position is 1 * 9!
100 choose 1
100
A teacher must select four members of the math club to participate in an upcoming competition. How many ways are there for her to make her selection if the club has 12 members? -- i.e., the number of ways of selecting an r-subset from a set of size n is n choose r express as n choose r
12 choose 4
How many 5-card hands are three of a kind? 3 of a kind example: 5H, 5S, 5C and 2D 2S -- 3 cards have same rank while other 2 have two different ranks
13*C(4, 3)*48*44 1) select the rank for the threes -- 13 possibilities 2) there are 4 cards with that rank (bc 4 suits), so we pick three -- C(4,3) 3) pick the 4th card, can't be the same rank as 3; removes 4 possibilties -- 52-4=48 4) pick the 5th card, can't be the same rank as 4th, remove 4 possibilities -- 48-4 = 44 ????
The math team has 6 girls and 4 boys. How many ways are there to select two competitors if they are both girls? answer numerically
15
A file will be replicated on 3 different computers in a distributed network of 15 computers. How many ways are there to select the locations for the file? express as n choose r
15 choose 3
Two dice are rolled. Enter the size of the set that corresponds to the event that one die is even and the other is odd.
18 even = 3 and odd = 3 multiplied = 9 switch order + 9 = 18 or, 3 for each, there are 6, so 18 { (1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6), (2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5) }.
An auto dealer has 7 different cars and 5 different trucks. How many ways are there to select three vehicles so that at least one truck is chosen?
185 1) total amt C(12, 3) = 220 2) non-trucks is C(7, 3) = 35 3) 220 - 35 = 185
An experiment consists of three consecutive flips of a coin. The outcome is the sequence of outcomes of the three flips. How many possible outcomes are there? The event that all three flips come out the same is E = {HHH, TTT}. The probability that all three flips come out the same is?
2/8 or 1/4 in uniform distribution p(E) = |E|/|S|
How many 8-bit strings have at least one 0?
255 1) We know that the set of 8 bit strings is 2^8 = 256 2) There is only 1 8-bit string which doesn't contain 0 -- 11111111 3) Therefore 256-1 = 255
(a) How many 8-bit strings have at least two consecutive 0's or two consecutive 1's? (b) How many 8-bit strings do not begin with 000?
255 1a) total amt of 8-bit strings is 2^8 = 256 2a) total amt of 8-bit strings with no consecutive zero's or one's -- 01010101 or 10101010 3a) 256 - 2 = 254 224 1b) total amt of 8-bit strings is 2^8 = 256 2b.) amt of strings that do begin with 000 -- 2^5 3b.) 2^8 - 2^5 = 224
Two dice are rolled. Enter the size of the set that corresponds to the event that the sum of the dice is 9.
4 The set of possible events is: { (3, 6), (4, 5), (5, 4), (6, 3) }.
If there are 13 possible ranks and 4 possible suits, how many cards in a standard set of cards are there?
4 * 13 = 52
There are 10 kids on the math team. Two kids will be selected from the team to compete in the state competition. How many ways are there to select the 2 competitors? answer numerically
45
A shop has 4 different shirts and 7 different jeans. How many ways are there to select two items so that *at least* one jeans is chosen?
49 1) total amt is C(11, 2) = 55 2) total amt of non-jeans 2-subsets is C(4, 2) = 6 3) C(11, 2) - C(4, 2) = 49
Two dice are rolled. Enter the size of the set that corresponds to the event that the sum of the dice is 8.
5 { (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) }.
In an experiment consisting of a roll of a red and blue die, what is the probability that the red die is one more than the blue die?
5/36 The event is the set E = {(1,2), (2,3), (3,4), (4,5), (5,6)}. The size of the sample space is 36.
Two dice are rolled. Enter the size of the set that corresponds to the event that the sum of the dice is 7.
6 The set of possible events is: { (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) }.
Consider a small example in which a subset of three colors is selected from the set Colors = {blue, green, orange, pink, red} Define a function mapping 3-permutations to 3-subsets. The function is defined by just removing the ordering, so (orange, pink, blue) and (blue, orange, pink) both map to the set {orange, blue, pink} Apply the k-1 rule to figure out the value of k -- how many permutations map to one 3-subset?
60/10 = 6 or 3! k-to-1 rule uses a k-to-1 correspondence to count the number of elements in the range by counting the number of elements in the domain and dividing by k. Suppose there is a k-to-1 correspondence from a finite set A to a finite set B. Then |B| = |A|/k. 60 permutations (domain) / 3! = 10 subsets (range)
Enter the size of the sample space when two 8-sided dice are rolled.
64 S={1,2,3,4,5,6,7,8}×{1,2,3,4,5,6,7,8}
A shop has 5 different shirts and 8 different jeans. How many ways are there to select two items so that at least one jeans is chosen?
68 1) total amt is C(13, 2) = 78 2) non-jeans is C(5, 2) = 10 3) 78 - 10 = 68
Dave swims three times in the week. How many ways are there to plan his workout schedule (i.e. which days he will swim) for a given week?
7 choose 3 A schedule for Dave's workouts consists of the set of three days he will swim (e.g., {Mon, Wed, Sat}). The order in which he selects the three days of the week does not affect the schedule. Therefore, Dave is selecting a subset of 3 days from the 7 days of the week.
A teacher selects 4 students from her class of 37 to work together on a project. How many ways are there for her to select the students?
C(37, 4)
How many different 5-card hands (5-subsets) are there in a standard playing deck?
C(52, 5)
The number of possible outcomes from dealing a 5-card hands?
C(52, 5)
How many five card hands have at least one club?
C(52, 5) - C(39, 5) 1) can calculate each term separately -- number of hands w one club, two, three, etc. 2) number of five card hands - number of five card hands with no clubs = five card hands with at least one club 3) C(52, 5) - C(39, 5)
A shop has 6 different shirts and 5 different jeans. How many ways are there to select 2 shirts?
C(6, 2)
100 choose 100
1
Define a bijection between 5-subsets of the set S = {1, 2, 3, 4, 5, 6, 7, 8} and 8-bit strings with exactly five 1's. A subset X of S with five elements maps on to a string x so that j ∈ X if and only if the jth bit of x is 1. 1) What string corresponds to the set {1, 3, 4, 5, 8}? 2) How many 8-bits strings have exactly five 1's? -- what is the number of ways of selecting an r-subset from a set of size n? 3) How many strings over the alphabet {a, b} have length 20 and exactly 8 a's?
1) 10111001 2) , since, it's a bijection with 5-subsets from set S, 8 choose 5 3) There are (20, 8) ways to select the 8 locations of the a's from among the 20 locations in the string. Then the b's go in the remaining 12 locations
Let S = {a, b, c}. 1) Is (b, a) a 2-permutation or a 2-subset from S? 2) Is {b, a} a 2-permutation or a 2-subset from S? 3) How many different 2-permutations from S are there? 4) How many different 2-subsets from S are there?
1) 2-permutation 2) 2-subset 3) 6; the 2-permutations from S are: (a, b), (b, a), (a, c), (c, a), (b, c), (c, b) -- order matters 4) 3; The 2-subsets from S are {a, b}, {a, c}, {b, c}.
1) How many binary strings of length nine have exactly four 1's? 2) How many strings of length eight over the set {0...9} have exactly three sevens?
1) Nine choose four 2) C(8,7) * 9^5
An auto dealer has 5 different cars and 4 different trucks. How many ways are there to select three vehicles so that at least one truck is chosen?
1) total amt is C(9, 3) = 84 2) non-trucks is C(5, 3) = 10 3) 84-10 = 74
Four people (John, Paul, George, and Ringo) are seated in a row on a bench. The number of ways to order the four people so that John is next to Paul is 12. How many ways are there to order the four people on the bench so that John is not next to Paul? answer numerically
1) total number of 4-permutations is 4! = 24 2) when they sit next to each other is 12 3) 24-12=12
A bit string contains 1's and 0's. How many different bit strings can be constructed given the restriction(s)? Length is 15. Starts with: 000
2^12
A bit string contains 1's and 0's. How many different bit strings can be constructed given the restriction(s)? Length is 22.
2^22
Suppose a coin is flipped three times. The outcome of the experiment is the sequence of outcomes from each flip. For example, HHH denotes the outcome in which the coin comes up heads in each flip. How many distinct outcomes are there?
2^3
How big is the set of 8 bit strings?
2^8
Two dice are rolled. Enter the size of the set that corresponds to the event that the sum of the dice is 4.
3 { (1, 3), (2, 2), (3, 1) }.
There are 20 members of a basketball team. 12 players are selected to travel to an away game. From the 12 players who will travel, the coach must select her starting line-up. She will select a player for each of the five positions: center, power forward, small forward, shooting guard and point guard. However, there are only three of the 12 players who can play center. Otherwise, there are no restrictions. How many ways are there for her to select the starting line-up??
3 * P(11, 4)
Consider a small example in which a subset of three colors is selected from the set Colors = {blue, green, orange, pink, red} How many permutations map to the selection {orange, blue, pink} ?
3!
A fair coin is flipped three times. Enter the probability that at least two consecutive flips come up the same.
3/4 The set of possible outcomes of the event is: { (T, T, T), (T, T, H), (H, T, T), (H, H, T), (T, H, H), (H, H, H) }. There are 6 possible outcomes. The sample space for the event that at least two consecutive flips come up the same is: S={H,T}×{H,T}×{H,T} There are 8 possible outcomes in the sample space. The probability for the event that at least two consecutive flips come up the same is: p(E)=|E|/|S|=6/8=3/4
A fair coin is flipped three times. Enter the probability that two flips come up tails and one comes up heads
3/8 The set of possible outcomes of the event is: { (T, T, H), (H, T, T), (T, H, T) }. There are 3 possible outcomes. The sample space for the event that two flips come up tails and one comes up heads is: S={H,T}×{H,T}×{H,T} There are 8 possible outcomes in the sample space. The probability for the event that two flips come up tails and one comes up heads is: p(E)=|E|/|S|=3/8
The math team has 6 girls and 4 boys. How many ways are there to select the two competitors so that at least one boy is chosen? answer numerically
30 1) total is C(10, 2) = 45 2) number of 2-subsets with only girls is C(6, 2) = 15 3) 45-15=30
There are 30 boys and 35 girls that try out for a chorus. The choir director will select 10 girls and 10 boys from the children trying out. How many ways are there for the choir director to make his selection?
30 choose 10 * 35 choose 10
A class of 30 students elects four students to serve on a student leadership council. The teacher tallies the votes and only reveals the names of the four students who received the most votes. How many different outcomes are there from the election process?
30 choose 4
How many different passwords are there that contain only digits and lower-case letters and satisfy the given restrictions? Length is 6 and the password must contain at least one digit and at least one letter.
36^6 - (26^6 + 10^6) 1) total amt is 36^6 2) non digits is 26^6 and non-letters is 10^6
How many different passwords are there that contain only digits and lower-case letters and satisfy the given restrictions? Length is 6 and the password must contain at least one digit.
36^6 - 26^6 1) total amt is 36^6 2) non-digits is 26^6)
How many length 8 strings over the alphabet {a, b, c} have at least one "a"?
3^8 - 2^8 1) total amount of strings is 3^8 2) amt of strings that don't have an a is 2^8 3) 3^8 - 2^8
how many suits of one rank?
4
Two dice are rolled. Enter the size of the set that corresponds to the event that both dice are odd
9 odd = 3 and odd = 3 = 9 The set of possible events is: { (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5) }.
A software team has 8 senior members and 12 junior members. The manager must select a set of 4 people to work on a project. How many selections have at least one junior member?
C(20, 4) - C(12, 4) 1) total amount of 4-subsets is C(20, 4) 2) amt with no junior members is C(12, 4)
A bit string contains 1's and 0's. How many different bit strings can be constructed given the restriction(s)? Length is 22. Has exactly seven 0's. Starts with: 11
C(20, 7)
A bit string contains 1's and 0's. How many different bit strings can be constructed given the restriction(s)? Length is 20. Has exactly twelve 0's.
C(20,12)
How many 5-card hands have no face cards? face cards (A, J Q, K)
C(36, 5) 1) 9 possible ranks 2) 4 of each rank, so 36 cards to choose from
An auto dealer has 4 different cars and 8 different trucks. How many ways are there to select two vehicles?
C(12, 2)
How many five card hands have exactly one club?
C(13, 1)*C(39, 4) 1) number of clubs in a deck = 13 2) number of non-clubs in a deck = 39 3) picking one club C(13, 1) 4) pick non-clubs C(39, 4)
How many 5-card hands have exactly 2 hearts?
C(13, 2)*C(39, 3) 1) there are 13 ranks with this suit -- chose 2 hearts 2) there are 52-13=39 cards that don't have this suit, so we choose 3 from 39
How many 5-card hands have 2 pairs? ex: 2H, 2C, & QD, QC & 9H
C(13, 2)*C(4,2)*C(4, 2)*44 1) Select ranks for the two pairs -- C(13,2) 2) select the cards for first pair -- C(4, 2) 3) select the cards for second pair -- C(4, 2) 4.) select the fifth card, cant be same suit as any pair -- 52-8 = 44
How many 5-card hands (5-subsets) have exactly 3 clubs?
C(13,3)*C(39,2) 1) there are 13 clubs total 2) select three that are clubs 3) select thee remaining two that are not clubs
A bit string contains 1's and 0's. How many different bit strings can be constructed given the restriction(s)? Length is 28. Has exactly seven 1's in the first half. Has exactly three 1's in the second half.
C(14, 7)*C(14, 3)
A bit string contains 1's and 0's. How many different bit strings can be constructed given the restriction(s)? Length is 16. Has exactly twelve 0's. express n choose r as C(n, r)
C(16, 12)
There are 20 members of a basketball team. The coach must select 12 players to travel to an away game. How many ways are there to select the players who will travel?
C(20, 12)
Consider a small example in which a subset of three colors is selected from the set Colors = {blue, green, orange, pink, red} How many 3-subsets of colors are there?
P(5,3)/3! = 5!/3!2! = 10 n!/(r!(n-r)!) = P(n/r)/r!
Consider a small example in which a subset of three colors is selected from the set Colors = {blue, green, orange, pink, red} How many 3-permutations are there from this set?
P(5,3) = 60
There are 20 members of a basketball team. 12 players are selected to travel to an away game. From 12 players, the coach must select her starting line-up. She will select a player for each of the five positions: center, power forward, small forward, shooting guard and point guard. How many ways are there for her to select the starting line-up?
P(12, 5)
The students in a class elect a president, vice president, secretary, and treasurer. There are 30 students in the class and no student can have more than one job. How many different outcomes are there from the election process?
P(30, 4) Select an order in which the class officers are elected: Pres, VP, Treas, Sec. The order in which the students are elected (in addition to which students are elected) is important because the order will determine which student gets which job. Therefore, a sequence of four kids are selected from the 30: (president, vice president, secretary, treasurer).
A teacher selects students from her class of 37 students to do 4 different jobs in the classroom: pick up homework, hand out permission slips, staple worksheets, and organize the classroom library. Each job is performed by exactly one student in the class and no student can get more than one job. How many ways are there for her to select students and assign them to the jobs?
P(37, 4)
After proving a bijection from the set of 5-bit strings with exactly two 1's to 2-subsets of {1,2,3,4,5}, we can find the size of the first set. How big is the set? -- what is the number of ways of selecting an r-subset from a set of size n? express as n choose r
Since the mapping is a bijection, (# of 5-bit strings with exactly 2 1's) = (# of 2-subsets of {1, 2, 3, 4, 5} = C(5, 2)
Consider a dishonest dice player who shows up to a game with a loaded die. The player's die is biased so that an outcome of 6 is twice as likely to occur as the other numbers. Define an experiment which is a roll of the single die *In the loaded die example above, what is the probability of the event that the number on the die is 5 or 6?*
The probability of the event {5, 6} is p(5) + p(6) = 1/7 + 2/7 = 3/7.
experiment in the case of red and blue die...
a procedure that results in one out of a number of possible *outcomes* the _______ is the rolling of the dice. the outcome of a roll of two dice is the number that shows up on the blue dice and the number that shows up on the red dice -- (5, 3) if the blue die is first and rolls a 5 and the red die is second and rolls a three
Consider as an example a distribution process in which a teacher is distributing a set of four prizes to the ten students in his class. Each student can get at most one prize. How many ways are there to distribute the prizes if: a) The prizes are all identical. b)The prizes are all different from each other.
a) 10 choose 4 b) P(10, 4) If the prizes are all different from each other, then the problem is asking about the number of 4-permutations from a set of 10 elements because the order in which prizes are distributed is important. If the prizes are all identical, then the problem is asking about the number of 4-subsets because the order in which prizes are distributed is not important.
How many different strings of length 12 containing exactly five a's can be chosen over the following alphabets? (a) The alphabet {a, b} (b) The alphabet {a, b, c}
a) 12 choose 5 b) (12 choose 5) * 2^7
Suppose a network has 40 computers of which 5 fail. (a) How many possibilities are there for the five that fail? (b) Suppose that 3 of the computers in the network have a copy of a particular file. How many sets of failures wipe out all the copies of the file? That is, how many 5-subsets contain the three computers that have the file?
a) 40 choose 5 b) 37 choose 2 because we have to choose the remaining spots fro computers that don't contain the file.
A search committee is formed to find a new software engineer. (a) If 100 applicants apply for the job, how many ways are there to select a subset of 9 for a short list? (b) If 6 of the 9 are selected for an interview, how many ways are there to pick the set of people who are interviewed? (You can assume that the short list is already decided). (c) Based on the interview, the committee will rank the top three candidates and submit the list to their boss who will make the final decision. (You can assume that the interviewees are already decided.) How many ways are there to select the list from the 6 interviewees?
a) C(100, 9) -- looking for a subset, order doesn't matter b) C(9, 6) c) C(6, 3)
120 pianists compete in a piano competition. (a) In the first round, 30 of the 120 are selected to go on to the next round. How many different outcomes are there for the first round? (b) In the second round, the judges select the first, second, third, fourth and fifth place winners of the competition from among the 30 pianists who advanced to the second round. How many outcomes are there for the second round of the competition?
a) C(120, 30) b) P(30, 5)
A group of five friends go to a restaurant for dinner. The restaurant offers 20 different main dishes. (a) Suppose that the group collectively orders five different dishes to share. The waiter just needs to place all five dishes in the center of the table. How many different possible orders are there for the group? (b) Suppose that each individual orders a main course. The waiter must remember who ordered which dish as part of the order. It's possible for more than one person to order the same dish. How many different possible orders are there for the group? (c) Suppose that each individual orders a main course. The waiter must remember who ordered which dish as part of the order. However the friends agree that no two people will order the same dish.
a) C(20, 5) b) 20^5 c) P(20, 5)
In the experiment of a 5-card hand, is the following an outcome or an event? {{4♣, 6♠, 7♥, Q♦, K♦}, {4♠, 9♥, 9♣, 9♦, K♦}}
event The given set has two elements, each of which is a set of 5 cards. Therefore, the set is an event with two outcomes. It's a subset of S.
when you see "at least"...?
count by complement
k-to-1 rule
k-to-1 rule uses a k-to-1 correspondence to count the number of elements in the range by counting the number of elements in the domain and dividing by k. Suppose there is a k-to-1 correspondence from a finite set A to a finite set B. Then |B| = |A|/k. domain/k = range
the number of ways of selecting an r-subset from a set of size n is
n choose r n!/r!(n−r)!
100 pianists compete in a piano competition. in the first round, 25 of the 100 contestants are selected to go on to the next round. how many possible outcomes are there to select the 25?
order doesn't matter so, 100 choose 25
100 pianists compete in a piano competition. in the first round, 25 of the 100 contestants are selected to go to the second round. in the third round, the judges select a first, second, third, fourth, and fifth place winner from the 25 who made it. How many outcomes are there for the third round?
order matters so P(25,5)
In the experiment of a 5-card hand, is the following an outcome or an event? {4♣, 6♠, 7♥, Q♦, K♦}
outcome Every outcome is a subset of 5 cards. Therefore the set of 5 cards given is an outcome.
sample space in the case of red and blue dice...
set of all possible outcomes S={1,2,3,4,5,6}×{1,2,3,4,5,6}
event in the case of the red and blue die... and (condition that the sum of the roles is exactly 8)
subset of the sample space (set of all possible outcomes) For example, the event E that the sum of the dice is exactly 8 is the following set: E={(2,6),(3,5),(4,4),(5,3),(6,2)}
The sample space of 5-card hands consists of...
the set of all 5-subsets of the 52 cards
how many clubs are there in a standard set of playing cards?
there are 13 club cards in a set of 52 and 39 non-clubs
In the example of a red and blue die that are thrown, define the event E to be that the number on both dice are multiples of 3. Which set corresponds to E?
{(3,3), (3,6), (6,3), (6,6)}
Consider a dishonest dice player who shows up to a game with a loaded die. The player's die is biased so that an outcome of 6 is twice as likely to occur as the other numbers. Define an experiment which is a roll of the single die The event that the die comes up an even number is _____. The probability of the event that the die comes up even is: ????????????? skip
{2, 4, 6} and 3/7 ???????????
The event Ejack that the hand has four jacks would be the set of all hands of the form ___________ . The number of outcomes in Ejack is ____
{J♠, J♣, J♥, J♦, *} where the "*" could be any of the other 48 cards that are not jacks. The number of outcomes in Ejack is 48.