Transcript Counting Techniques

How many ways could…?
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There are 9 boys and 8 girls in the student
council at Hermitage High School. How many
ways could a single student be selected to
hold a single office?
There are
◦ 9+8=17 ways of making the selection
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The events of selecting a boy and selecting a
girl are called Mutually Exclusive or Disjoint.
This is because a person cannot be both a
boy and a girl.
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How many ways can a die and a coin fall
together?
First make two Blanks
_ _
 Then write the number of possibilities for the die
and the coin on the blanks.
6
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2
Now multiply those numbers together to get
12.
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Think back to the problem with Hermitage’s
student council. If both one boy and one girl
are to be selected to hold two different
offices on the council, how many ways can
the offices be filled?
Remember there are 9 boys and 8 girls and to
make your two blanks!
9 × 8 = 72
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Suppose, from the 17 students in the student
council, how many ways can you choose a
president, secretary, and treasurer? One
person cannot hold more than one position.
If the president is elected first, how many
options are there?
17
How many options are left for the secretary if
that position is elected next?
16
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How many options are there for the treasurer
if that position is chosen third?
15
Now use the multiplication principle to find
your final answer.
17 × 16 × 15 = 4,080
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If there are 5 students waiting to get in line to
get on a bus, how many orders can the first
two students get on the bus?
5 × 4 = 20
Type equation here.
There are 20 different orders for the first two
students to get on the bus.
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The formula for a permutation is…
P
𝑛!
𝑛−𝑟 !
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n r=
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5 students, the first two would be 5 2
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Note that 6! represents 6 × 5 × 4 × 3 × 2 × 1
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P
TI-84 press the number for n, math, PRB, the
second one on the list, then the number for r.
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A Combination is used to describe a selection
of several objects where the order they were
chosen does not matter.
Think of the first two students entering the
bus. If you only need to know who those first
two students were and not the order. For
example Brian then Darren would be the
same as Darren then Brian.
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We knew there were 20 orders the first two
student could enter the bus.
For each pair of students there were two
orders.
How many combinations of the first two
students can be made?
20
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2
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= 10
There are 10 combinations of the first two
students on the bus.
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nCr=
𝑛!
𝑛−𝑚 !𝑚!
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5C2=
5!
5−2 !2!
=
5!
3!2!
=
5×4×3×2×1
3×2×1×2×1
=
5×4
2×1
= 10
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Permutations: order matters
Combinations: order does not, think groups
If there are 8 students and 8 tests versions, how
many ways can the tests be given out?
8!=40,320
How many two letter “words” can be made from
the word FLOWERS?
7P2=42
9 boys and 8 girls in student council. How many
four person comities can consist of 3 boys and 1
girl?
9C3*8C1=84*8=672