Permutations

P(6,6)

10. There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?

P(26, 4)

A password consists of four different letters of the alphabet. How many different possible passwords are there?

* 26 * 26 * 10 * 10 * 10 = 676000

A password consists of two letters of the alphabet followed by three digits chosen from 0 to 9. Repeats are allowed. How many different possible passwords are there?

* 62^6 + 62^7 + 62^8

A password must contain from 6 to 8 case sensitive alpha numeric characters How many possible passwords are there?

* P(8!)

An encyclopedia has eight volumes. In how many ways can the eight volumes be replaced on the shelf?

* 12 + 20 - 4

Flight Example 12 flights that begin in Portland 20 flights that end in houston 4 flights that begin in portland and end in houston How many flights are there that either begin in Portland or end in Houston?

* 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 = 127 - 1 (remove empty string) = 126

12. How many bit strings are there of length six or less, not counting the empty string?

* C(12, 0) + C(12, 1) + C(12, 2) + C(12, 3) = 1 + 12 + 66 + 220 + 299

12. How many bit strings of length 12 contain b) at most three 1s?

* Find total possible combinations and subtract off the combinations we do not waht 0 , 1, and 2. 2^12 - C(12, 0) - C(12, 1) - C(12, 2) = 4096 - 1 - 12 - 66 = 4,017

12. How many bit strings of length 12 contain c) at least three 1s?

* C(12, 6) = 12 * 11 * 10 * 9 * 8 * 7 / 6! = 665,280 / 720 = 924

12. How many bit strings of length 12 contain d) an equal number of 0s and 1s?

* C(12, 3) = 12!/3!(12-3)! = 12!/3!9! = 12 * 11 * 10 / 3 * 2 * 1 = 1320/6 = 220

12. How many bit strings of length 12 contain a) exactly three 1s?

* n = 5 r = 20 C(5+20-1, 20) = C(24, 20) = 24!20!15!= 10,626

12. How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a piggy bank contain if it has 20 coins in it?

* n = 4 things to choose from r = 17 items to select C(4+17-1, 17) = C(20, 17) = 20!17!3!= 1140

14. How many solutions are there to the equation x1 + x2 + x3 + x4 = 17, where x1, x2, x3, and x4 are nonnegative integers?

* 11: {1, 10}, {2, 9}, {3, 8}, {4, 7}, {5, 6}. Let's start with drawing six integers. If six integers are selected from the first 10 integers, by the Pigoenhole Principle at least ⌈6/5 ⌉ = 2 numbers select 7 numbers no repitition ; the 6 and 7 pick will make for two pairs

14. a. Show that if seven integers are selected from the first 10 positive integers, there must be at least two pairs of these integers with the sum 11.

* no only one pair will be picked

14. a. Show that if seven integers are selected from the first 10 positive integers, there must be at least two pairs of these integers with the sum 11. b. Is the conclusion in part (a) true if six integers are selected rather than seven?

* 26^4 - 25^4 = 66351 OR you can also use the concept if if there is only one x there are 4 positions C(4,1) * 25 * 25 * 25 then add this to 2 xs C(4,2) * 25 * 25 add this to three positions C(4,3) * 25 ADD TO ALL POISITINSO XXXX IS ONE 62500 + 3750 + 100 + 1 = 66351

16 . How many strings are there of four lowercase letters that have the letter x in them?

* P(10,10) + P(11,6) = P(10!) * P(11,6) = 10! * 11!/6!(5!) = 3,628,800 * 332,640 = 1,207,084,032,000

24. How many ways are there for 10 women and six men to stand in a line so that no two men stand next to each other? [Hint: First position the women and then consider possible positions for the men.]

* 10 * 10 * 10 * 5 = 50000

26. How many strings of four decimal digits b) end with an even digit?

* 10 * 8 * 7 * 6 P(10, 4)

26. How many strings of four decimal digits a) do not contain the same digit twice?

* x999 9x99 99x9 999x x = 0-8 so there are only 9 digits to choose from This would be 9 * 4 = 36

26. How many strings of four decimal digits c) have exactly three digits that are 9s?

* Subtract the one case where there are no women on the field. C(13, 10) - 1 = 13! /10!(13-10)! - 1 = 13!/10!*3! - 1 = 286 -1 = 285

26. Thirteen people on a softball team show up for a game. c) Of the 13 people who show up, three are women. How many ways are there to choose 10 players to take the field if at least one of these players must be a woman?

* C(13, 10)

26. Thirteen people on a softball team show up for a game. a) How many ways are there to choose 10 players to take the field?

* P(13, 10)

26. Thirteen people on a softball team show up for a game. b) How many ways are there to assign the 10 positions by selecting players from the 13 people who show up?

* P(26, 4)

A password consists of four different letters of the alphabet. How many different possible passwords are there?

* 10^3 * 26^3 * 2

28. How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits?

* containers 99,999,999 objects 100,000,000

32. Show that if there are 100,000,000 wage earners in the United States who earn less than 1,000,000 dollars (but at least a penny), then there are two who earned exactly the same amount of money, to the penny, last year.

* 6 objects 5 contianers = 6/5 = 2

36. A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.

* 5 n /2 = 3

4. A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. a) How many balls must she select to be sure of having at least three balls of the same color?

* 13

4. A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. b) How many balls must she select to be sure of having at least three blue balls?

* 2 ^5 + 2^4 - 2^2 ( the one that starts with two 0s and ends with three 1s only has 2 bits in the middle )

48. How many bit strings of length seven either begin with two 0s or end with three 1s?

* 38 + 23 - 7

52. Every student in a discrete mathematics class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 38 computer science majors (including joint majors), 23 mathematics majors (including joint majors), and 7 joint majors?

* n= 3 r= 5 C(5+3-1, 5)

6. How many ways are there to select five unordered elements from a set with three elements when repetition is allowed?

* The remainders will always be 0,1,2,3...n-1 these are the containers d + 1 group of numbers will be the objects there will always be one more number than containers

6. Let d be a positive integer. Show that among any group of d + 1 (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by d.

* 26 * 25 * 24= 15600 p(26,3)

8. How many different three-letter initials with none of the letters repeated can people have?

* n = 21 r = 12 C(12 + 21-1, 12) = C(32, 12)

8. How many different ways are there to choose a dozen donuts from the 21 varieties at a donut shop?

* C(30,6)

A group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission (assuming that all crew members have the same job?

* Perm with rep 3^5

Consider a bag of jelly beans that has 100 red, 100 yellow, and 100 green jelly beans. b) How many color sequences can you get by drawing 5 beans from the bag?

* n = 2 (no more yellow) r = 8 C(2+8-1, 8) = C(10,8)

Consider a bag of jelly beans that has 100 red, 100 yellow, and 100 green jelly beans. c) How many color combinations of 10 beans have exactly two yellow beans?

* n = 3 still can choose yellow, r = 8 times times choosing c(3+8-1, 8) = C(10,8) = 45

Consider a bag of jelly beans that has 100 red, 100 yellow, and 100 green jelly beans. d) How many color combinations of 10 beans have at least two yellow beans?

* n = 3 r = 10 C(10+3-1, 10) = C(12,10)

Consider a bag of jelly beans that has 100 red, 100 yellow, and 100 green jelly beans. a) How many color combinations can you get by drawing 10 beans from the bag?

* 4 containers result s3 9 balls

Consider a bag that contains 10 blue balls, 10 red balls, 20 green balls, and 20 yellow balls. Now suppose that we take n balls from the bag at random. a) How large does n need to be in order to guarantee that we get at least 3 balls of the same color?

* n = 53

Consider a bag that contains 10 blue balls, 10 red balls, 20 green balls, and 20 yellow balls. Now suppose that we take n balls from the bag at random. a) How large does n need to be in order to guarantee that we get at least 3 blue balls?

* C(10, 5)

Consider a group of 10 people. b) How many ways are there to select a group of 5 people from the full group.

* P(10, 5)

Consider a group of 10 people. a) How many ways are there to form a line of 5 people from the group?

* containers 12 objects 100 100/12 = 8 .33 = 9

Consider a party with 100 people. What can we say about the number of people that were born in the same month?

* 365 container 366 objects yes pigeonhole

Consider a party with 366 people. Are there at least two people with the same birthday assuming a standard year?

2^5

Count the number of bit strings of length 5 00000, 10000,..., 11111

5 x 6 x 3

Count the number of possible levels Supposed room designer: 5 different styles 6 color schemes 3 different luxury levels

*10 * 9 * 8 = 720 P(10,3)

How many permutations of 3 different digits are there, chosen from the ten digits 0 to 9 inclusive?

* P(6, 6) = DEFGH 5 and ABC 1

How many permutations of the letters ABCDEFGH contain the string ABC ?

* C(52, 5)

How many poker hands of five cards can be dealt from a standard deck of 52 cards?

* 6^10 - 5^10 =

How many strings of length 10 constructed from characters in the set {a, b, c, d, e, f} contain at least one occurrence of the letter 'a'. (As usual the strings do not need to contain all 6 characters and can contain characters multiple times.)

* 26^r

How many strings of length r can be formed from the uppercase letters of the English alphabet?

* P(5,5) bride groom count as one

How many ways are there to line up a wedding party of 6 people assume that the groom must always be lined up directly after the bride?

* C(52, 47)

How many ways are there to select 47 cards from a standard deck of 52 cards

* P(100,3)

How many ways are there to select a first-prize winner, a second-prize winner, and a third-prize winner from 100 different people who have entered a contest?

* n = 7 r = 5 C(7+5-1, 7)

How many ways are there to select five bills from a cash box containing $1 bills, $2 bills, $5 bills, $10 bills, $20 bills, $50 bills, and $100 bills? Assume that the order in which the bills are chosen does not matter, that the bills of each denomination are indistinguishable, and that there are at least five bills of each type.

* C(10, 5)

How many ways are there to select five players from a 10-member tennis team to make a trip to a match at another school?

* n - 3 r : 4 C(4+3-1, 4)

How many ways are there to select four pieces of fruit from a bowl containing apples, oranges, and pears if the order in which the pieces are selected does not matter, only the type of fruit and not the individual piece matters, and there are at least four pieces of each type of fruit in the bowl?

* n = 10 r = 5 C(10, 5)

In how many ways can a committee of 5 be chosen from 10 people?

* P(5!)

In how many ways can we arrange all five of these students in a line for a picture?

* n = 5 r = 3 P(5,3)

In how many ways can we select three students from a group of five students to stand in line for a picture?

* P(5,3)

In how many ways can we select three students from a group of five students to stand in line for a picture?

* 62^5 - 52^5

Passwords must contain exactly 5 case sensitive alphanumeric characters and must have at least one digit. How many possible passwords are there?

* containers - 0,1,2,3 possible reminders

Show that among any set of 5 positive integers, there are 2 with the same remainder when divided by 4.

* containers 0,1,2,4 objects 9 = 9/4 = 2.25 = 3

Show that among any set of 9 positive integers, there are 3 with the same remainder when divided by 4.

* n= 4 r = 6 C(4+6-1, 6)

Suppose that a cookie shop has four different kinds of cookies. How many different ways can six cookies be chosen? Assume that only the type of cookie, and not the individual cookies or the order in which they are chosen, matters.

* P(7!) = 5040

Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities?

* C(9,3) * C(10,4)

Suppose that there are 9 faculty members in the mathematics department and 11 in the computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members from the mathematics department and four from the computer science department?

* P(6,6)

There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?

* C(13, 10)

Thirteen people on a softball team show up for a game. How many ways are there to choose 10 players to take the field

* 12 + 5 + 3

Travel with 12 plane options, 5 bus options, and 3 train options. How many possible trip options are there.

* P (3, 3) do not factor in a and e to n or r

a) How many strings are there that use each character in the set {a, b, c, d, e} exactly once (i.e. permutations of {a,b,c,d,e}) and that start with an 'a' and end with an 'e'?

* p(4,4) de count as one n and r The sequence "de" can show up in the first through fourth positions, and there are 3 characters left to permute where we have to use all three so 4 * P(3,3) = 4 * 3! / (3 - 3)! = 4 * 6 = 24

b) How many strings are there that use each character in the set {a, b, c, d, e} exactly once (i.e. permutations of {a,b,c,d,e}) and that contain the sequence 'de' somewhere in the string?