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```In SuperLotto Plus, a California state lottery game, you
select five distinct numbers from 1 to 47, and one MEGA
number from 1 to 27, hoping that your selection will
match a random list selected by lottery officials.
(a) How many different sets of six numbers can you
select?
In SuperLotto Plus, a California state lottery game, you
select five distinct numbers from 1 to 47, and one MEGA
number from 1 to 27, hoping that your selection will
match a random list selected by lottery officials.
(a) How many different sets of six numbers can you
select?
Answer: C(47,5)xC(27,1)
= 1,533,939 x 27
= 41,416,353
In SuperLotto Plus, a California state lottery game, you
select five distinct numbers from 1 to 47, and one MEGA
number from 1 to 27, hoping that your selection will
match a random list selected by lottery officials.
Eileen Burke always includes her age and her
husband’s age as two of the first five numbers in her
SuperLotto Plus selections. How many ways can she
complete her list of six numbers?
In SuperLotto Plus, a California state lottery game, you
select five distinct numbers from 1 to 47, and one MEGA
number from 1 to 27, hoping that your selection will
match a random list selected by lottery officials.
Eileen Burke always includes her age and her
husband’s age as two of the first five numbers in her
SuperLotto Plus selections. How many ways can she
complete her list of six numbers?
Answer: C(45,3)xC(27,1)
= 14,190x 27
= 383,130
Drawing Cards - How many cards must be
drawn (without replacement) from a
standard deck of 52 to guarantee drawing
two cards of the same suit?
Drawing Cards - How many cards must be
drawn (without replacement) from a
standard deck of 52 to guarantee drawing
two cards of the same suit?
Answer: 5
Drawing Cards - How many cards must be
drawn (without replacement) from a
standard deck of 52 to guarantee drawing
three cards of the same suit?
Drawing Cards - How many cards must be
drawn (without replacement) from a
standard deck of 52 to guarantee drawing
three cards of the same suit?
Answer: 9
How many different 5-card poker
hands would contain only cards of a
single suit?
How many different 5-card poker
hands would contain only cards of a
single suit?
Answer: 4 x C(13,5)
= 5148
Subject identification numbers in a certain
scientific research project consist of three
letters followed by three digits and then
three more letters. Assume repetitions are
not allowed within any of the three groups,
but letters in the first group of three may
occur also in the last group of three. How
many distinct identification numbers are
possible?
Subject identification numbers in a certain
scientific research project consist of three
letters followed by three digits and then
three more letters. Assume repetitions are
not allowed within any of the three groups,
but letters in the first group of three may
occur also in the last group of three. How
many distinct identification numbers are
possible?
Answer: P(26,3) x P(10,3) x P(26,3)
= 175,219,200,000
Radio stations in the United States have
call letters that begin with K or W (for west
or east of the Mississippi River,
respectively). Some have three call letters,
such as WBZ in Boston, WLS in Chicago,
and KGO in San Francisco. Assuming no
repetition of letters, how many three-letter
sets of call letters are possible?
Radio stations in the United States have
call letters that begin with K or W (for west
or east of the Mississippi River,
respectively). Some have three call letters,
such as WBZ in Boston, WLS in Chicago,
and KGO in San Francisco. Assuming no
repetition of letters, how many three-letter
sets of call letters are possible?
Answer: 2 x P(25,2)
= 1200
Most stations that were licensed after
1927 have four call letters starting with
K or W, such as WXYZ in Detroit or
KRLD in Dallas. Assuming no
repetitions, how many four-letter sets
are possible?
Most stations that were licensed after
1927 have four call letters starting with
K or W, such as WXYZ in Detroit or
KRLD in Dallas. Assuming no
repetitions, how many four-letter sets
are possible?
Answer: 2 x P(25,3)
= 27,600
Each team in an eight-team basketball
league is scheduled to play each other
team three times. How many games
will be played altogether?
Each team in an eight-team basketball
league is scheduled to play each other
team three times. How many games
will be played altogether?
Answer: 3 x C(8,2)
= 84
The Coyotes, a youth league baseball
team, have seven pitchers, who only pitch,
and twelve other players, all of whom can
play any position other than pitcher. For
Saturday’s game, the coach has not yet
determined which nine players to use nor
what the batting order will be, except that
the pitcher will bat last. How many different
batting orders may occur?
The Coyotes, a youth league baseball
team, have seven pitchers, who only pitch,
and twelve other players, all of whom can
play any position other than pitcher. For
Saturday’s game, the coach has not yet
determined which nine players to use nor
what the batting order will be, except that
the pitcher will bat last. How many different
batting orders may occur?
Answer: 7 x P(12,8)
= 139,708,800
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that a girl always performs first?
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that a girl always performs first?
Answer: 8 x 14!
= 697,426,329,600
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that a girl always performs first and a
boy always performs second?
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that a girl always performs first and a
boy always performs second?
Answer: 8 x 7 x 13!
= 348,713,164,800
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that Lisa always performs first and
Doug always performs second?
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that Lisa always performs first and
Doug always performs second?
Answer: 1x13!x1
= 6,227,020,800
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that the entire program will alternate
between girls and boys?
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that the entire program will alternate
between girls and boys?
Answer: 8! x 7!
= 203,212,800
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that the first, eighth and fifteenth
performers must be girls?
A music class of eight girls and seven
boys is having a recital. If each
member is to perform once, how many
ways can the program be arranged so
that the first, eighth and fifteenth
performers must be girls?
Answer: 8 x 7 x 6 x 12!
= 160,944,537,600
Carole begins each day by reading
from one of seven inspirational books.
How many ways can she choose the
books for one week if the selection is
done by placing the book back on the
bookshelf after she reads it?
Carole begins each day by reading
from one of seven inspirational books.
How many ways can she choose the
books for one week if the selection is
done by placing the book back on the
bookshelf after she reads it?
Answer: 77
= 823,543
Carole begins each day by reading
from one of seven inspirational books.
How many ways can she choose the
books for one week if the selection is
done by choosing a different book
each day?
Carole begins each day by reading
from one of seven inspirational books.
How many ways can she choose the
books for one week if the selection is
done by choosing a different book
each day?
Answer: 7!
= 5,040
How many of the possible 5-card
hands from a standard 52-card deck
would consist of four clubs and one
non-club?
How many of the possible 5-card
hands from a standard 52-card deck
would consist of four clubs and one
non-club?
Answer: C(13,4) x 39
= 27,885
How many of the possible 5-card
hands from a standard 52-card deck
would consist of two face cards and
three non-face cards?
How many of the possible 5-card
hands from a standard 52-card deck
would consist of two face cards and
three non-face cards?
Answer: C(12, 2) x C(40, 3)
= 652, 080
How many of the possible 5-card
hands from a standard 52-card deck
would consist of two red cards two
clubs and a spade?
How many of the possible 5-card
hands from a standard 52-card deck
would consist of two red cards two
clubs and a spade?
Answer: C(26, 2) x C(13, 2) x 13
= 329,550
In how many ways could twenty-five
people be divided into five groups
containing, respectively, three, four,
five, six, and seven people?
In how many ways could twenty-five
people be divided into five groups
containing, respectively, three, four,
five, six, and seven people?
Answer: C(25,3) x C(22,4) x C(18,5) x C(13,6)
= 2.474 x 1014
How many different threenumber “combinations” are
possible on a combination
lock having 40 numbers on
its dial? (Hint: “Combination”
is a misleading name for
these locks since repetitions
are allowed and also order
makes a difference.)
How many different threenumber “combinations” are
possible on a combination
lock having 40 numbers on
its dial? (Hint: “Combination”
is a misleading name for
these locks since repetitions
are allowed and also order
makes a difference.)
Answer : 403
= 64,000
Michael Grant, his wife and son, and four
additional friends are driving, in two
vehicles, to the seashore. If all seven
people are available to drive, how many
ways can the two drivers be selected?
(Everyone would like to drive the sports car,
so it is important which driver gets which
car.)
Michael Grant, his wife and son, and four
additional friends are driving, in two
vehicles, to the seashore. If all seven
people are available to drive, how many
ways can the two drivers be selected?
(Everyone would like to drive the sports car,
so it is important which driver gets which
car.)
Answer: P(7,2)
= 210
At the race track, you win the “daily
double” by purchasing a ticket and
selecting the winners of both of two
specified races. If there are six and
eight horses running in the first and
second races, respectively, how many
tickets must you purchase to
guarantee a winning selection?
At the race track, you win the “daily
double” by purchasing a ticket and
selecting the winners of both of two
specified races. If there are six and
eight horses running in the first and
second races, respectively, how many
tickets must you purchase to
guarantee a winning selection?
Answer: 6 x 8
= 48
Many race tracks offer a “trifecta” race.
You win by selecting the correct first-,
second-, and third-place finishers. If
eight horses are entered, how many
tickets must you purchase to
guarantee that one of them will be a
trifecta winner?
Many race tracks offer a “trifecta” race.
You win by selecting the correct first-,
second-, and third-place finishers. If
eight horses are entered, how many
tickets must you purchase to
guarantee that one of them will be a
trifecta winner?
Answer: P(8,3)
= 336
Because of his good work, Jeff Hubbard gets a
contract to build homes on three additional blocks
in the subdivision, with six homes on each block.
He decides to build nine deluxe homes on these
three blocks: two on the first block, three on the
second, and four on the third. The remaining nine
homes will be standard. Altogether on the threeblock stretch, how many different choices does
Jeff have for positioning the eighteen homes?
(Hint: Consider the three blocks separately and
use the fundamental counting principle.)
Because of his good work, Jeff Hubbard gets a
contract to build homes on three additional blocks
in the subdivision, with six homes on each block.
He decides to build nine deluxe homes on these
three blocks: two on the first block, three on the
second, and four on the third. The remaining nine
homes will be standard. Altogether on the threeblock stretch, how many different choices does
Jeff have for positioning the eighteen homes?
(Hint: Consider the three blocks separately and
use the fundamental counting principle.)
Answer: C(6,2) x C(6,3) x C(6,4)
= 4500
How many six-digit counting numbers
can be formed using all six digits 4, 5,
6, 7, 8, and 9?
How many six-digit counting numbers
can be formed using all six digits 4, 5,
6, 7, 8, and 9?
Answer: 6! or P(6,6)
= 720
A professor teaches a class of 60
students and another class of 40
students. Five percent of the students
in each class are to receive a grade of
A. How many different ways can the A
grades be distributed?
A professor teaches a class of 60
students and another class of 40
students. Five percent of the students
in each class are to receive a grade of
A. How many different ways can the A
grades be distributed?
Answer: C(60,3) x C(40,2)
= 26,691,600
How many counting numbers have
four distinct nonzero digits such that
the sum of the four digits is 12?
How many counting numbers have
four distinct nonzero digits such that
the sum of the four digits is 12?
Answer: 2 x 4!
= 48
Only 2 ways to get a sum of 12
Using 1,2,3 and 6 or 1,2,4 and 5
A computer company will screen a
shipment of 30 processors by testing a
random sample of five of them. How
many different samples are possible?
A computer company will screen a
shipment of 30 processors by testing a
random sample of five of them. How
many different samples are possible?
Answer: C(30,5)
= 142,506
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