Counting
Unit 1
Counting Activity 1
1. Let H be heads and T be tails. How many
different results can we get from 1 flip of the
coin?
Counting Activity 2
1. How many different results can we get from two
flips? How about three?
2. We call each flip an event. An event is a discrete
activity, sometimes each event takes place at
different times or sometimes the events can
take place at the same time. In the case of our
coin flips, each flip is considered an event.
3. The flip of a coin is also called a discrete event
because we can’t have a half or a quarter of a
coin flip.
Counting Activity 3
1. Using a tree diagram to help with counting?
a. For each event we will use a a branch of the tree.
b. Each possible result will be a limb on the branch.
c. We count the total number of limbs on the final
branch of the tree.
2. Suppose that you own a sandwich shop, you
have two kinds of bread (wheat and white),
three kinds of cheese(cheddar, swiss, mozarella),
and four kinds of meat(roast beef, turkey,
chicken, ham). How many different kinds of
sandwiches could you make?
Counting Activity 3
Roast Beef
Chicken
Cheddar
Turkey
Ham
Roast Beef
Chicken
White
Swiss
Turkey
Ham
Roast Beef
Chicken
Mozarella
Turkey
Ham
Sandwiches
Roast Beef
Chicken
Cheddar
Turkey
Ham
Roast Beef
Wheat
Swiss
Chicken
Turkey
Ham
Roast Beef
Chicken
Mozarella
Turkey
Ham
Counting Activity 4
• Suppose that you and four friends wanted to
play some golf. You need to choose an order
to play in, 1 through 5. How many different
orders can you play in? (Construct a tree
diagram to show the different pairings.)
The Multiplication Principle
• Take a look at the tree diagram for Counting Activity 3.
– We have two different kinds of bread, three different kinds
of cheese and four different kinds of meat, and 24
different kinds of sandwiches.
– That is: 2 x 3 x 4= 24.
• The multiplication principle tells us that if we have a
certain number of elements of a group and we want to
figure out the total number of groupings that we can
get with these elements, then we multiply the number
of each event by the number of the other events. That
is called the multiplication principle. Kind of tricky to
write down, but simple to put into practice!
Counting Activity 5
1. Suppose that you flip a coin 5 times, how many different
results can you get for the group of 5 flips?
2. A straight in poker is when you have five consecutive
cards, for example, two, three, four, five, six, of any suit (w
call this a six high straight). There are four suits in a deck
of cards (hearts, spades, clubs, and diamonds). How
many ways can you get a ten high straight (six, seven,
eight, nine and ten of any suit)?
3. In poker, a three of a kind is when you get three of the
same number card (for example, the four of spades, the
four of clubs, and the four of the diamonds). How many
ways can you get three of a kind with kings?
Factorials
• Suppose there are five people who need to be
placed in order from 1 to 5? How many ways can
they be ordered?
• According to the multiplicative principle, we can
count the total number by multiplying:
5 x 4 x 3 x 2 x 1=120
• When you begin at a particular number and then
multiply it by the next smallest number and then
stop at 1, this is called a factorial and is written as
5!.
Factorials
• We write the factorial as 5!=120.
• You can find the factorial key in your
calculator. On the scientific calculators, this is
actually a key that you can shift to.
• On the TI-83, 84, and 84-Plus, you press the
number (5)
– MATH-PROB-the factorial key looks like an
exclamation point.
Repetitions
• How many different ways can we order the
letters in the word Mississippi?
• We can’t just use 11! because we have four
choices for i, four choices for s, two choices for
p and one m. We need to eliminate the
counting for the redundant letters. We need
to divide out the redundant letters. How
many ways can we order the i’s? The s’s? The
p’s?
Repetitions
• For the total number of words we can make
from Mississippi, we would use:
11!
= 34, 650
4!* 4!* 2!
Permutations
• When we want to find out how many groups or results
we can get when we have a specific number of
elements and when each of the elements has a specific
role or order then we use permutations.
• Permutations utilize the multiplicative principal in that
we can think of the permutation as being the
multiplication of individual events in descending order.
• For example suppose that we have five people that we
want to put into a club’s group of offices, president,
vice-president, and secretary. How many groups of
president, vice-president and secretary can we make
with those five people?
Permutations
• Suppose that we have 5 people (Sally, Bobby,
Juan, Theresa, and Luz) for each office we can
have one person. For the president, we can
choose from any of the five, for the second we
can then choose four, and finally for the
secretary we have three. Using the
multiplicative principle, we get:
5 x 4 x 3=60
Permutations
• You can use your calculator in order to find
permutations.
– On a standard scientific calculator. You will find on
one of the shift keys, a key that says nPr or nPx. This
is the permutation key. n is the number of elements
(or people that you can choose from) and r or x is the
number that you need in each group.
• So for this problem we would press 5 nPr 3.
– For a TI-83, 84, or 84-Plus MATH-PROB-and the
button that says nPr.
– In the same way as the scientific calculator, 5 nPr 3
and enter.
Combinations
• Suppose that we wanted to figure out how
many groups of elements we could get from a
larger group if each of the elements did not
have a set role or order.
• For example, suppose that you had 10
different kinds of jellybeans, how many
different groups of three can you get from
those 10 jellybeans?
Combinations
• Suppose that we tried to use the same technique as
permutations. For the first jellybean we have 10 choices,
for the second we have 9, and the third we have eight.
Suppose that the first jellybean was cinnamon, the second
jellybean was grape, and the third was licorice. This is one
group. However, in this case, if licorice was the first,
cinnamon was the second, and grape was the third, then
this group is still the exact same group. In the case of our
club committee, Juan as the president, Bobby as the vicepresident, and Sally is the secretary is a different group
from Bobby as the president, Sally as the vice-president,
and Juan as the secretary. In the case of the jellybeans we
can’t use the permutation because we would be doublecounting, we use a function called a combination.
Combination
• Unlike the permutation, we need to actually divide out
the redundant groups. In the case of our jellybeans, how
many redundant groups would we have from the
cinnamon, licorice, and grape jellybeans?
• Using the multiplicative principle and factorials, we have:
3 x 2 x 1= 3!=6.
• So the total number of three jellybean groups that we
can get from ten different jellybeans is:
10 * 9 *8 720
=
=120
3!
6
Combinations
• We can use the calculator to find a
combination.
• For a scientific calculator, there is a shift key
nCr that is similar to the permutation. n is the
number of elements that we are choosing
from and r is the number in each group.
• On the graphing calculator, TI-83, TI-84, TI-84
Plus, use MATH-PROB-nCr. So for the
jellybean problem, we would use 10 nCr 3.
Choosing when to use Combinations
or Permutations
• We say that we use a Permutation when order
matters. When we use a Combination we say
that order does not matter.
• Order mattering means that each element in a
particular group has a particular role or order.
Like the example where each member of the
committee was either the president, vicepresident, and secretary.
• Order not mattering is like the jellybean example
where the three different jellybeans can be
placed into any order.
Sample Spaces
• The set of all objects that we are counting or
considering for a particular event or set of
events is called the sample space. Counting
the number of elements in the sample space
is something that is used frequently in
probability.
Review for Homework
• New Terms
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Event
Discrete
Sample Space
Combination
Permutation
• Key ideas and techniques
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Developing Tree Diagrams
The Multiplicative Principle
Factorials
Combinations and Permutations