Deck 6: A: Counting

Question 1

How many words have no vowels?

Answer 1

Question 2

How many words begin with A or B and end with A or B?

Answer 2

Question 3

How many words have exactly one vowel?

Answer 3

Question 4

consider all bit strings of length 12.
How many have exactly four 1's?

Question 5

How many words begin with R and end with T?

Question 6

How many words begin with AAB in some order?

Question 7

consider all bit strings of length 12.
How many begin with 110?

Question 8

How many words begin with A or end with B?

Question 9

consider all bit strings of length 12.
How many begin with 11 or end with 10?

Question 10

How many words begin with a vowel and end with a vowel?

Question 11

How many words begin with A or B or end with A or B?

Question 12

How many words begin with a vowel or end with a vowel?

Question 13

consider all bit strings of length 12.
How many have exactly four 1's and none of these 1's are adjacent to each other?

Question 14

How many permutations of the seven letters A, B, C, D, E, F, G are there?

Question 15

How many permutations of the seven letters A, B, C, D, E, F, G have E in the first position?

Question 16

How many words end with the letter T?

Question 17

In questions 1-12 suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed.
How many words are there?

Question 18

consider all bit strings of length 12.
How many begin with 11 and end with 10?

Question 19

Find the number of words of length eight of distinct letters of the alphabet so that the words do not have both A and B in them.

Question 20

How many words begin with A or B?

Question 21

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if Hal or Ida (but not both) are in the picture?

Question 22

How many 8-element DNA sequences contain exactly two of the four bases?

Question 23

How many permutations of the seven letters A, B, C, D, E, F, G have A immediately to the left of E ?

Question 24

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if neither Ed nor Fran are in the picture?

Question 25

How many truth tables are possible for compound propositions with the five variables p, q, r, s, t ?

Question 26

let A be the set of all bit strings of length 10.
How many bit strings of length 10 are there?

Question 27

How many 8-element DNA sequences start with C and end with C?

Question 28

How many permutations of the seven letters A, B, C, D, E, F, G do not have the vowels next to each other?

Question 29

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if both Ed and Gail are in the picture?

Question 30

How many permutations of the seven letters A, B, C, D, E, F, G have E in one of the first two positions?

Question 31

How many permutations of the seven letters A, B, C, D, E, F, G neither begin nor end with A?

Question 32

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if Dot is on the left end and Ed is on the right end?

Question 33

In a technician's box there are 400 VLSI chips, 12 of which are faulty. How many ways are there to pick two chips, so that one is a working chip and the other is faulty? (Assume that no chips are identical.)

Question 34

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if Ed and Gail are in the picture, standing next to each other?

Question 35

How many 5-element DNA sequences use all four bases, A, T, C, and G?

Question 36

How many permutations of the seven letters A, B, C, D, E, F, G have the two vowels before the five conso-

Question 37

let A be the set of all bit strings of length 10.
How many bit strings of length 10 begin with 1101?

Question 38

How many permutations of the seven letters A, B, C, D, E, F, G do not have vowels on the ends?

Question 39

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if Ann and Ben are in the picture, but not standing next to each other?

Question 40

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them
stand in a row for a picture.
In how many ways can this be done if Ben is to be in the picture?

Question 41

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all
30 books are different.
In how many ways can you get a bunch of three history books and seven novels to give to a friend?

Question 42

let A be the set of all bit strings of length 10.
How many bit strings of length 10 have exactly six 0's?

Question 43

let A be the set of all bit strings of length 10.
How many bit strings of length 10 have equal numbers of 0's and 1's?

Question 44

let A be the set of all bit strings of length 10.
How many bit strings of length 10 have more 0's than 1's?

Question 45

a club with 20 women and 17 men needs to form a committee of size six.
How many committees are possible if the committee must have three women and three men?

Question 46

A class consists of 20 sophomores and 15 freshmen. The class needs to form a committee of size five. (a) How many committees are possible? (b) How many committees are possible if the committee must have three sophomores and two freshmen?

Question 47

Suppose =10 and =4 . Find the number of 1-1 functions

Question 48

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all
30 books are different.
In how many ways can you put the 30 books in a row on a shelf?

Question 49

A club with 20 women and 17 men needs to choose three different members to be president, vice president, (a) In how many ways is this possible? (b) In how many ways is this possible if women will be chosen as president and vice president and a man as treasurer?

Question 50

A class consists of 20 sophomores and 15 freshmen. The club needs to choose four different members to be president, vice president, secretary, and treasurer. (a) In how many ways is this possible? (b) In how many ways is this possible if sophomores will be chosen as president and treasurer and freshmen as vice president and secretary?

Question 51

a club with 20 women and 17 men needs to form a committee of size six.
How many committees are possible if the committee must consist of all women or all men?

Question 52

Suppose =4 and =10 . Find the number of functions

Question 53

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all
30 books are different.
In how many ways can you put the 30 books in a row on a shelf if the five math books are to be grouped together, but there are no restrictions on the placement of the other books?

Question 54

let A be the set of all strings of decimal digits of length five. For example, 00312 and 19483
are strings in A.
How many strings in A begin with 774?

Question 55

Suppose =4 and =10 . Find the number of 1-1 functions

Question 56

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all
30 books are different.
In how many ways can you get a bunch of four books to give to a friend?

Question 57

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all
30 books are different.
In how many ways can you put the 30 books in a row on a shelf if the novels are on the left, the math books are in the middle, and the history books are on the right?

Question 58

let A be the set of all strings of decimal digits of length five. For example, 00312 and 19483
are strings in A.
Find |A|.

Question 59

a club with 20 women and 17 men needs to form a committee of size six.
How many committees are possible?

Question 60

a club with 20 women and 17 men needs to form a committee of size six.
How many committees are possible if the committee must have at least two men?

Question 61

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin and end with T.

Question 62

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors.
In how many ways can you get a committee of 4 freshmen and 3 sophomores?

Question 63

How many ways are there to select 6 students from a class of 25 to hold six different executive positions on

Question 64

How many subsets with an odd number of elements does a set with 10 elements have?

Question 65

How many subsets with more than two elements does a set with 100 elements have?

Question 66

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8.

Question 67

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin and end with the same letter.

Question 68

Make up a word problem in good English whose answer is

Question 69

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have exactly one B.

Question 70

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors.
In how many ways can you put all students in a line so that the freshmen are first, the sophomores are in the middle, and the juniors are at the end?

Question 71

let A be the set of all strings of decimal digits of length five. For example, 00312 and 19483
are strings in A.
How many strings in A have exactly one 5?

Question 72

Each user has a password 6 characters long where each character is an uppercase letter, a lowercase letter, or a digit. Each password must contain at least one digit. How long will it take to check every possible character combination, if each check takes one unit of time.

Question 73

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin with L or end with R.

Question 74

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors.
In how many ways can you put all 30 in a line?

Question 75

Make up a word problem in good English whose answer is

Question 76

How many ways are there to select 6 students from a class of 25 to serve on a committee?

Question 77

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

Question 78

let A be the set of all strings of decimal digits of length five. For example, 00312 and 19483
are strings in A.
How many strings in A have exactly three 5s?

Question 79

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have at least one C.

Question 80

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors.
In how many ways can you get a committee of 7?