MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
He still has 4
choices of pants
𝑃1
𝑇1
𝑃2
𝑃3
𝑃4
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
He still has 4
choices of pants
𝑇1
And for each
𝑃1
𝑃2
𝑃3
𝑃4
choice of
pants, he has 6
choices of
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6
jackets
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
He still has 4
choices of pants
𝑇1
So, the designer has 24
different outfits that
include top, 𝑇1 .
And for each
𝑃1
𝑃2
𝑃3
𝑃4
choice of
pants, he has 6
choices of
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6
jackets
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
Tops = {𝑇1 , 𝑇2 , 𝑇3 , 𝑇4 , 𝑇5 },Pants = {𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 },
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
But the designer does not have to stick with Top, 𝑇1 .
MATH 110 Sec 12.1 Intro
𝑇
𝑇 to
X1 Counting Practice Exercises
2
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
If he decided to use Top, 𝑇2 instead, there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
𝑇
𝑇 to
X1 Counting Practice Exercises
2
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
and the designer has 24 different outfits that include top, 𝑇2
If he decided to use Top, 𝑇2 instead, there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
𝑇3 𝑇 to
𝑇
X1 Counting Practice Exercises
2
X
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So,
So, the
the designer
designer has
has 24
24 different
different outfits
outfits that
that include
include top,
top, 𝑇𝑇11.
and the designer has 24 different outfits that include top, 𝑇2
If he decided to use Top, 𝑇3 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
𝑇
𝑇33 𝑇2to
X1 Counting Practice Exercises
X2
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
and the designer has 24 different outfits that include top, 𝑇2
and the designer has 24 different outfits that include top, 𝑇3
If he decided to use Top, 𝑇3 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
𝑇
X3 𝑇 to𝑇X1 𝑇Counting Practice Exercises
2
X
4
A designer designed 5 different
tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So,
So, the
the designer
designer has
has 24
24 different
different outfits
outfits that
that include
include top,
top, 𝑇𝑇11.
and the designer has 24 different outfits that include top, 𝑇2
and the designer has 24 different outfits that include top, 𝑇3
If he decided to use Top, 𝑇4 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
𝑇
𝑇X33 𝑇2to
X1 𝑇Counting Practice Exercises
X2 4
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
and the designer has 24 different outfits that include top, 𝑇2
and the designer has 24 different outfits that include top, 𝑇3
and the designer has 24 different outfits that include top, 𝑇4
If he decided to use Top, 𝑇4 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
Practice Exercises
𝑇
𝑇5
X3 𝑇 to𝑇X1 𝑇Counting
2 X4
X
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So,
So, the
the designer
designer has
has 24
24 different
different outfits
outfits that
that include
include top,
top, 𝑇𝑇11.
and the designer has 24 different outfits that include top, 𝑇2
and the designer has 24 different outfits that include top, 𝑇3
and the designer has 24 different outfits that include top, 𝑇4
Finally, if he decided to use Top, 𝑇5 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
Practice Exercises
𝑇
𝑇5
𝑇X33 𝑇2to
X1 𝑇Counting
X2 X4
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
Suppose the designer has already decided to use the Top, 𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
and the designer has 24 different outfits that include top, 𝑇2
and the designer has 24 different outfits that include top, 𝑇3
and the designer has 24 different outfits that include top, 𝑇4
and the designer has 24 different outfits that include top, 𝑇5
Finally, if he decided to use Top, 𝑇5 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro
Practice Exercises
𝑇
𝑇5
𝑇X33 𝑇2to
X1 𝑇Counting
X2 X4
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
𝑃3 model possibly
𝑃4
1 pants & 1𝑃jacket)
2
top, 1 pair 𝑃of
can the
wear?
Tops = {𝑇 , 𝑇 , 𝑇 , 𝑇 , 𝑇 },Pants = {𝑃 , 𝑃 , 𝑃 , 𝑃 },
𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽61 𝐽1 2𝐽2 𝐽33 𝐽4 𝐽45 𝐽65 𝐽1 𝐽2 𝐽3 𝐽4 𝐽5 𝐽6 𝐽11 𝐽2 2𝐽3 𝐽43 𝐽5 𝐽46
Jacket = {𝐽1 , 𝐽2 , 𝐽3 , 𝐽4 , 𝐽5 , 𝐽6 }
So, all together,
there would
24 x 5 decided
= 120 different
outfits.
Suppose
the designer
has be
already
to usepossible
the Top,
𝑇1 .
So, the designer has 24 different outfits that include top, 𝑇1
and the designer has 24 different outfits that include top, 𝑇2
and the designer has 24 different outfits that include top, 𝑇3
and the designer has 24 different outfits that include top, 𝑇4
and the designer has 24 different outfits that include top, 𝑇5
Finally, if he decided to use Top, 𝑇5 , there would be another 24 different outfits.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
Note: Because we also covered the section on the
Fundamental Counting Principle, we could actually use that to
answer this question much more efficiently.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
# ways to choose top
x # ways to choose pants x # ways to choose jacket =
possible
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
5
# ways to choose top
x # ways to choose pants x # ways to choose jacket =
possible
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
5
# ways to choose top
x
4
x # ways to choose pants x # ways to choose jacket =
possible
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
5
# ways to choose top
x
4
x
6
x # ways to choose pants x # ways to choose jacket =
possible
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
5
# ways to choose top
x
4
x
6
=
x # ways to choose pants x # ways to choose jacket =
120
possible
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
A designer designed 5 different tops, 4 different pants and
6 different jackets. How many different outfits (consisting of 1
top, 1 pair of pants & 1 jacket) can the model possibly wear?
So the model could possibly wear 120 different outfits.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
you get the same answer as
A task composed of aNotice
seriesthat
of sub-tasks
in which the first
before without
havingthe
to draw
a tree
sub-task can be performed
in a ways,
second
in diagram.
b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
5
# ways to choose top
x
4
x
6
=
x # ways to choose pants x # ways to choose jacket =
120
possible
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
Because a tree diagram here would be very large, we will
once again take advantage of the fact that we have already
covered the section on the Fundamental Counting Principle.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
Here we will be choosing from 4 possible letters
(U, V, E, A) and 3 possible numbers (8, 3, 7).
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
x
2
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
x
2
x
1
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
x
2
x
1
x
3
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
x
2
x
1
x
3
x
2
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
x
2
x
1
x
3
x
2
x
1
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
4
x
3
x
2
x
1
x
3
x
2
x
1
= 144
#waysto
#waysto
#waysto
#waysto
# ways
# ways
# ways
choose x choose x choose x choose x choose x choose x choose = TOT
1st letter
2nd letter
3rd letter
4th letter
1st #
2nd #
3rd #
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
An eyewitness to a crime said that the license plate of the
getaway car began with the four letters U, V, E and A (but he
couldn’t remember the order). The rest of the plate had the
numbers 8, 3 and 7 but, again, he could not remember the
order. How many license plates fit the eyewitness description?
144 license plates fit the eyewitness description.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
Two couples (Adam/Brenda and
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
Two couples (Adam/Brenda and
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
To make it easier to see how to keep
the 2 males apart, let’s replace the
male names with 𝑴𝟏 & 𝑴𝟐 and the
female names with 𝑭𝟏 and 𝑭𝟐 .
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
To make it easier to see how to keep
the 2 males apart, let’s replace the
male names with 𝑴𝟏 & 𝑴𝟐 and the
female names with 𝑭𝟏 and 𝑭𝟐 .
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Our task is to list every possible way
that the couples could be seated
without the men sitting together.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Suppose 𝑀1 sits in the first seat.
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Suppose 𝑀1 sits in the first seat.
We can’t seat 𝑀2 in seat 2 but we can
seat either 𝐹1 or 𝐹2 there.
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Suppose 𝑀1 sits in the first seat.
We can’t seat 𝑀2 in seat 2 but we can
seat either 𝐹1 or 𝐹2 there.
𝐹1
𝑀1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀1 sits in the first seat and 𝐹1 sits in
the second seat, then either of the
two remaining (𝑀2 or 𝐹2 ) can be next.
𝐹1
𝑀1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀1 sits in the first seat and 𝐹1 sits in
the second seat, then either of the
two remaining (𝑀2 or 𝐹2 ) can be next.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀1 sits in the first seat and 𝐹1 sits in
the second seat, then either of the
two remaining (𝑀2 or 𝐹2 ) can be next.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now let’s move to the case in which
𝑀1 sits in seat 1 and 𝐹2 sits in seat 2.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀1 sits in the first seat and 𝐹2 sits in
the second seat, then either of the
two remaining (𝑀2 or 𝐹1 ) can be next.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀1 sits in the first seat and 𝐹2 sits in
the second seat, then either of the
two remaining (𝑀2 or 𝐹1 ) can be next.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀1 sits in the first seat and 𝐹2 sits in
the second seat, then either of the
two remaining (𝑀2 or 𝐹1 ) can be next.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝑀2 sits in the first seat.
𝐹1
𝑀1
𝐹2
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝑀2 sits in the first seat.
We can’t seat 𝑀1 in seat 2 but we can
seat either 𝐹1 or 𝐹2 there.
𝐹1
𝑀1
𝐹2
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝑀2 sits in the first seat.
We can’t seat 𝑀1 in seat 2 but we can
seat either 𝐹1 or 𝐹2 there.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀2 sits in the first seat and 𝐹1 sits in
the second seat, then either of the
two remaining (𝑀1 or 𝐹2 ) can be next.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀2 sits in the first seat and 𝐹1 sits in
the second seat, then either of the
two remaining (𝑀1 or 𝐹2 ) can be next.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀2 sits in the first seat and 𝐹1 sits in
the second seat, then either of the
two remaining (𝑀1 or 𝐹2 ) can be next.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now let’s move to the case in which
𝑀2 sits in seat 1 and 𝐹2 sits in seat 2.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀2 sits in the first seat and 𝐹2 sits in
the second seat, then either of the
two remaining (𝑀1 or 𝐹1 ) can be next.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀2 sits in the first seat and 𝐹2 sits in
the second seat, then either of the
two remaining (𝑀1 or 𝐹1 ) can be next.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝑀2 sits in the first seat and 𝐹2 sits in
the second seat, then either of the
two remaining (𝑀1 or 𝐹1 ) can be next.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Once 3 people are seated, the only
person yet to be seated must sit in
the fourth seat.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝐹1 sits in the first seat.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝐹1 sits in the first seat.
This forces us to seat a male next.
(Otherwise, the 2 males would have
to sit next to each other in seats 3 & 4.)
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝐹1 sits in the first seat.
This forces us to seat a male next.
(Otherwise, the 2 males would have
to sit next to each other in seats 3 & 4.)
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀1 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀2 in seat 4 to keep the
males separated.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀1 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀2 in seat 4 to keep the
males separated.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀1 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀2 in seat 4 to keep the
males separated.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀2 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀1 in seat 4 to keep the
males separated.
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀2 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀1 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀2 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀1 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝐹2 sits in the first seat.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝐹2 sits in the first seat.
This forces us to seat a male next.
(Otherwise, the 2 males would have
to sit next to each other in seats 3 & 4.)
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
Now suppose 𝐹2 sits in the first seat.
This forces us to seat a male next.
(Otherwise, the 2 males would have
to sit next to each other in seats 3 & 4.)
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
𝑀1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹2 sits in the first seat and 𝑀1 sits in
the second seat, then 𝐹1 must sit in
seat 3 and 𝑀2 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
𝑀1
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹2 sits in the first seat and 𝑀1 sits in
the second seat, then 𝐹1 must sit in
seat 3 and 𝑀2 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝑀1
𝐹1
𝑀2
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹2 sits in the first seat and 𝑀1 sits in
the second seat, then 𝐹1 must sit in
seat 3 and 𝑀2 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝑀1
𝐹1
𝑀2
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀2 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀1 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝑀1
𝐹1
𝑀2
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
𝑀2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀2 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀1 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝑀1
𝐹1
𝑀2
𝑀2
𝐹1
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
If 𝐹1 sits in the first seat and 𝑀2 sits in
the second seat, then 𝐹2 must sit in
seat 3 and 𝑀1 in seat 4 to keep the
males separated.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2
𝑀2
𝐹1
𝑀2
𝐹2
𝑀1
𝐹1
𝑀1
𝑀2
𝑀2
𝐹2
𝑀1
𝑀1
𝐹1
𝑀2
𝑀2
𝐹1
𝑀1
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2 1
𝑀2 2
𝐹1 3
𝑀2 4
𝐹2 5
𝑀1 6
𝐹1 7
𝑀1 8
𝑀2 9
𝑀2
𝐹2
𝑀1 10
𝑀1
𝐹1
𝑀2 11
𝑀2
𝐹1
𝑀1 12
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
𝑴𝟏
𝑭𝟏
Two
couples (Adam/Brenda and
𝑴 𝟐 𝑭𝟐
Carl/Darlene) bought tickets to a
musical. In how many ways can the
couples be seated if the men do not
sit together?
So, there are 12 different ways these
two couples can be seated in which
the men do not sit together.
𝑀1
𝑀2
𝐹2
𝑀2
𝐹1
𝑀1
𝐹2
𝑀1
𝐹1
𝐹2
𝐹2 1
𝑀2 2
𝐹1 3
𝑀2 4
𝐹2 5
𝑀1 6
𝐹1 7
𝑀1 8
𝑀2 9
𝑀2
𝐹2
𝑀1 10
𝑀1
𝐹1
𝑀2 11
𝑀2
𝐹1
𝑀1 12
𝐹1
𝑀1
𝐹2
𝐹1
𝑀2
𝐹2
𝐹1
𝐹2
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
Although we could draw a tree diagram here, we will once
again take advantage of the fact that we have already
covered the section on the Fundamental Counting Principle.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
Here we will be choosing from 2 possible
answers (T or F) for each of the 4 questions.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
x
2
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
x
2
x
2
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
x
2
x
2
x
2
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
x
2
x
2
x
2
=
16
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
x
2
x
2
x
2
=
16
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many different ways are there to answer all 4 questions?
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
2
x
2
x
2
x
2
=
16
# ways to
answer Q1
x
# ways to
answer Q2
x
# ways to
answer Q3
x
# ways to
answer Q4
=
TOT
16 different ways to answer all four T/F questions.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get no questions wrong?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get no questions wrong?
Remember that from the previous problem, there are
16 different ways to answer all four T/F questions.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get no questions wrong?
Remember that from the previous problem, there are
16 different ways to answer all four T/F questions.
T
F
T
T
F
F
T
T
F
T
F
F
T
F
T F T F T F T F T F T F T F T F
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get no questions wrong?
Remember that from the previous problem, there are
16 different ways to answer all four T/F questions.
One of those 16 is the “key” to
T
F
the quiz and unless you match
the “key” exactly, you will miss
T
F
T
F
at least one problem.
T
F
T
F
T
F
T
F
T F T F T F T F T F T F T F T F
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get no questions wrong?
Remember that from the previous problem, there are
16 different ways to answer all four T/F questions.
One of those 16 is the “key” to
T
F
the quiz and unless you match
the “key” exactly, you will miss
T
F
T
F
at least one problem.
So, there is only 1 way to get
none of the 4 questions wrong.
T
F
T
F
T
F
T
F
T F T F T F T F T F T F T F T F
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
It is easy to see that the only way
R
X
to get exactly one wrong is to get:
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
It is easy to see that the only way
R
X
to get exactly one wrong is to get:
just the 1st one wrong (XRRR) or
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
It is easy to see that the only way
R
X
to get exactly one wrong is to get:
just the 1st one wrong (XRRR) or
R
X
R
X
just the 2nd one wrong (RXRR) or
R
X
R
X
R
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
It is easy to see that the only way
R
X
to get exactly one wrong is to get:
just the 1st one wrong (XRRR) or
R
X
R
X
just the 2nd one wrong (RXRR) or
just the 3rd one wrong (RRXR) or
R
X
R
X
R
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
It is easy to see that the only way
R
X
to get exactly one wrong is to get:
just the 1st one wrong (XRRR) or
R
X
R
X
just the 2nd one wrong (RXRR) or
just the 3rd one wrong (RRXR) or
R X
R X
R X
R X
just the 4th one wrong (RRRX)
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get one question wrong?
Here it might be easier to think of a quiz that has already
been graded. Let ‘R’ be ‘right’ and ‘X’ be ‘wrong’.
It is easy to see that the only way
R
X
to get exactly one wrong is to get:
just the 1st one wrong (XRRR) or
R
X
R
X
just the 2nd one wrong (RXRR) or
just the 3rd one wrong (RRXR) or
R X
R X
R X
R X
just the 4th one wrong (RRRX)
So, there are 4 ways to get one wrong.
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
1. RRXX
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
1. RRXX
R
X
2. RXRX
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
1. RRXX
R
X
2. RXRX
3. RXXR
R
X
R
X
R
X
R
X
R
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
1. RRXX
R
X
2. RXRX
3. RXXR
R
X
R
X
4. XRRX
R
X
R
X
R
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
1. RRXX
R
X
2. RXRX
3. RXXR
R
X
R
X
4. XRRX
R X
R X
R X
R X
5. XRXR
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
We already have the tree diagram, so perhaps the easiest
thing to do is count the number of branches with 2 X’s.
1. RRXX
R
X
2. RXRX
3. RXXR
R
X
R
X
4. XRRX
R X
R X
R X
R X
5. XRXR
6. XXRR
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
How many ways are there to get two questions wrong?
1.
2.
3.
4.
5.
6.
RRXX
RXRX
RXXR
XRRX
XRXR
XXRR
R
So, there are 6
ways to get two
wrong.
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
1. RRRR
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
1. RRRR
2. RRRX
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
1. RRRR
2. RRRX
3. RRXR
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
1. RRRR
2. RRRX
3. RRXR
4. RXRR
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
1. RRRR
2. RRRX
3. RRXR
4. RXRR
5. XRRR
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
Note that ‘3 or more correct’ means ‘3 correct’ or ‘4 correct’.
Now we are counting the
branches with either 3 or 4 R’s.
1. RRRR
2. RRRX
So, you have 5
3. RRXR
chances out of 16
4. RXRR
to get three or
5. XRRR
more correct.
R
X
R
R
X
X
R
R
X
R
X
X
R
X
R X R X R X R X R X R X R X R X
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
There is actually a faster way to solve this problem if you
realize that getting 3 correct is the same as getting 1 wrong
and getting 4 correct is the same as getting none wrong.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
There is actually a faster way to solve this problem if you
realize that getting 3 correct is the same as getting 1 wrong
and getting 4 correct is the same as getting none wrong.
The reason this helps is because we already found that there
were 4 ways of getting one wrong and that there was just 1
way of getting no questions wrong.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you are taking a 4 question T/F quiz:
If you randomly guess at each answer, what are your chances
of getting 3 or more correct?
There is actually a faster way to solve this problem if you
realize that getting 3 correct is the same as getting 1 wrong
and getting 4 correct is the same as getting none wrong.
The reason this helps is because we already found that there
were 4 ways of getting one wrong and that there was just 1
way of getting no questions wrong.
So, there are 4 + 1 = 5 ways to get three or more correct.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
Although you could solve this with a tree diagram (where no
branch can have all 3 flavors and each branch must have 2 flavors),
we will instead use the Fundamental Counting Principle (FCP).
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
# ways to choose
1st flavor
x
# ways to choose 2nd
flavor
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
# ways to choose 2nd
flavor
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
this apoint,
we need
to notecone
that we
arevanilla,
not allowed
to
If youAtbuy
3-scoop
ice cream
with
chocolate
and
have 3 of
same flavor
nor how
can we
havedifferent
one of each
flavor.
strawberry
asthe
possible
flavors,
many
cones
are
possible
if matter
your cone
2 of can
the only
flavors?
(Flavors
can be
So, no
what,has
theonly
3rd flavor
be chosen
from
repeated
and
two cones
are considered
different
if the
amongor2 not
of the
3 available
flavors.
This is because
if the first
2
flavors
are match,
the same
but occur
a different
order.
flavors
we can’t
chooseinthat
flavor again
and if the 2
flavors
not match, we can’t
choose thePRINCIPLE
3rd flavor. (FCP)
THEdo
FUNDAMENTAL
COUNTING
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
this apoint,
we need
to notecone
that we
arevanilla,
not allowed
to
If youAtbuy
3-scoop
ice cream
with
chocolate
and
have 3 of
same flavor
nor how
can we
havedifferent
one of each
flavor.
strawberry
asthe
possible
flavors,
many
cones
are
possible
if matter
your cone
2 of can
the only
flavors?
(Flavors
can be
So, no
what,has
theonly
3rd flavor
be chosen
from
repeated
and
two cones
are considered
different
if the
amongor2 not
of the
3 available
flavors.
This is because
if the first
2
flavors
are match,
the same
but occur
a different
order.
flavors
we can’t
chooseinthat
flavor again
and if the 2
flavors
not match, we can’t
choose thePRINCIPLE
3rd flavor. (FCP)
THEdo
FUNDAMENTAL
COUNTING
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
this apoint,
we need
to notecone
that we
arevanilla,
not allowed
to
If youAtbuy
3-scoop
ice cream
with
chocolate
and
have 3 of
same flavor
nor how
can we
havedifferent
one of each
flavor.
strawberry
asthe
possible
flavors,
many
cones
are
possible
if matter
your cone
2 of can
the only
flavors?
(Flavors
can be
So, no
what,has
theonly
3rd flavor
be chosen
from
repeated
and
two cones
are considered
different
if the
amongor2 not
of the
3 available
flavors.
This is because
if the first
2
flavors
are match,
the same
but occur
a different
order.
flavors
we can’t
chooseinthat
flavor again
and if the 2
flavors
not match, we can’t
choose thePRINCIPLE
3rd flavor. (FCP)
THEdo
FUNDAMENTAL
COUNTING
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
2
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
2
=
18
x
# ways to choose 3rd
flavor
=
TOTAL
MATH 110 Sec 12.1 Intro to Counting Practice Exercises
If you buy a 3-scoop ice cream cone with vanilla, chocolate and
strawberry as possible flavors, how many different cones are
possible if your cone has only 2 of the flavors? (Flavors can be
repeated or not and two cones are considered different if the
flavors are the same but occur in a different order.
THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)
A task composed of a series of sub-tasks in which the first
sub-task can be performed in a ways, the second in b ways,
the third in c ways, and so on, can be done in a x b x c x ... ways.
3
# ways to choose
1st flavor
x
3
x
# ways to choose 2nd
flavor
x
2
=
18
x
# ways to choose 3rd
flavor
=
TOTAL
So, there are 18 different cones satisfying the given conditions.