Deck 6: Counting

Question 1

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with A or end with B?

Answer 1

Question 2

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with a vowel and end with a vowel?

Answer 2

Question 3

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words have no vowels?

Answer 3

Question 4

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words have exactly one vowel?

Question 5

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with A or B?

Question 6

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

Question 7

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with R and end with T?

Question 8

In questions 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 9

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

Question 10

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with a vowel or end with a vowel?

Question 11

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words end with the letter T?

Question 12

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with A or B and end with A or B?

Question 13

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

Question 14

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

Question 15

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

Question 16

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 17

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

Question 18

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with A or B or end with A or B?

Question 19

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

Question 20

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters
allowed.
How many words begin with AAB in some order?

Question 21

In questions 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 22

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

Question 23

In questions 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 24

In questions 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 25

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

Question 26

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

Question 28

In questions 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 29

In questions 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 30

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

Question 31

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

Question 32

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 33

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

Question 34

How many 8-element DNA sequences contain exactly four C's?

Question 35

In questions 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 36

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

Question 37

In questions 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 38

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

Question 39

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

Question 40

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

Question 41

In questions 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 42

In questions 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 43

In questions 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?
In questions 45-48 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

Question 44

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

Question 45

In questions 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 46

In how many ways can you put the 30 books in a row on a shelf?

Question 47

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

Question 48

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

Question 49

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

Question 50

In how many ways can you get a bunch of three history books and seven novels to give to a friend?

Question 51

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 52

A.

Question 53

In questions 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 54

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

Question 55

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

Question 56

A club with 20 women and 17 men needs to choose three different members to be president, vice president,
and treasurer.
(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 57

In questions 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 58

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 59

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 60

In how many ways can you get a bunch of four books to give to a friend?

Question 61

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

Question 62

In questions 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?

Question 63

Make up a word problem in good English whose answer is

Question 64

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain the number 5.

Question 65

In questions 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 66

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

Question 67

Make up a word problem in good English whose answer is

Question 68

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 69

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

Question 70

In questions 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 71

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

Question 72

In questions 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 73

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 74

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 75

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

Question 76

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

Question 77

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

Question 78

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

Question 79

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

Question 80

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