Permutations
E. Jones
Permutations vs.
Combinations
Both are ways to count the possibilities
The difference between them is whether order
matters or not
Consider a poker hand:
Is that the same hand as:
A♦, 5♥, 7♣, 10♠, K♠
K♠, 10♠, 7♣, 5♥, A♦
Does the order the cards are handed out matter?
If yes, then we are dealing with permutations
If no, then we are dealing with combinations
Permutation or Combination?
Choosing two dogs from a litter of two males
and three females…
Permutation or Combination?
Choosing two dogs from a litter of two males
and three females…
Combination
Permutation or Combination?
A simple melody formed by playing the notes
on 8 different piano keys…
Permutation or Combination?
A simple melody formed by playing the notes
on 8 different piano keys…
Permutation
Permutation or Combination?
A selection of nine muffins from a shelf of
twenty-three…
Permutation or Combination?
A selection of nine muffins from a shelf of
twenty-three…
Combination
Permutation or Combination?
Choosing a six digit alphanumeric password to
access a website…
Permutation or Combination?
Choosing a six digit alphanumeric password to
access a website…
Permutation
Permutation or Combination?
The ways 6 race horses can place first through
third in a race…
Permutation or Combination?
The ways 6 race horses can place first through
third in a race…
Permutation
Permutation or Combination?
Dinner guests seated around a table…
Permutation or Combination?
Dinner guests seated around a table…
permutation
What is a permutation?
It is an arrangement of
a set of objects in a
particular order.
Order matters
For example…
In how many ways can Matt,
Raquel, and Jordan line up?
3 ⋅ 2 ⋅1 = 6
6 different ways
Next example…
Suppose you have to write thank-you
notes to 7 people. In how many
ways can you choose the order in
which to write the notes?
There are 7 choices for the 1st, 6
nd
choices for the 2 …
7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 5040
Is there an easier way?
7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 5040
This product can be written as 7!
You read this as seven
factorial.
factorial.
Can you use a calculator?
Find the n! or x! button on your
calculator.
Find 6! =720
Let’s put it all together…
Many CD players can shuffle the order in
which songs play. Your favorite CD has
9 songs.
In how many different orders can
the songs be played?
9! =362,880 ways WOW!
PERMUTATION NOTATION
A class of 15 students must choose a
president and a vice-president.
15
Fifteen objects
15P2=
P
2
Groups of two
15·14 =210
PERMUTATION NOTATION
The expression nPr
stands for the number
of permutations of n
objects r at a time.
Find the following…
At a horse show with 20 horses,
ribbons are given for 1st, 2nd, 3rd,
th
and 4 place. How many different
arrangements of four winning
horses are possible?
20P4 = 20·19·18·17 = 116,280
One more example…
10P3
= 10·9·8 = 720
You will study combinations later in high
school. Right now it is only important for you
to be able to tell the difference and be able to
calculate a permutation.
Let’s practice more…
Combination or Permutation?
A pattern of different widths of bars and spaces
for a bar code…
Let’s practice more…
Combination or Permutation?
A pattern of different widths of bars and spaces
for a bar code…
permutation
Permutation or Combination?
Selecting two yellow marbles out of a sack of
yellow and blue marbles…
Permutation or Combination?
Selecting two yellow marbles out of a sack of
yellow and blue marbles…
combination
Permutation or Combination?
Placing one can of each of 15 different types of
soup along a store shelf
Permutation or Combination?
Placing one can of each of 15 different types of
soup along a store shelf
permutation
Permutation or Combination?
Selecting four candles from a box of ten …
Permutation or Combination?
Selecting four candles from a box of ten …
combination
Permutation or Combination?
The placement of the top ten finishers in a
school’s spelling bee…
Permutation or Combination?
The placement of the top ten finishers in a
school’s spelling bee…
permutation
Permutation or Combination?
Choosing two colors of paint out of twenty to
paint the walls and trim of a bedroom…
Permutation or Combination?
Choosing two colors of paint out of twenty to
paint the walls and trim of a bedroom…
permutation