Chapter 6.4: Permutations and Combinations
Example:
Suppose six friends wish to line up for a photo. How many
ways can this be done?
Using the rule of product, there are six tasks to complete: who is the
first person from the left, who is the second, the third, fourth, fifth, and
sixth.
There are six choices for the first person. Then, there will be five choices
for the second person, four for the third person, and so on.
6
×
5
×
4
×
3
×
2
×
1
=
720
—–
—–
—–
—–
—–
—–
1st
2nd
3rd
4th
5th
6th
There are 720 different arrangements.
Example:
Suppose only five friends wish to line up for a photo. How
many different arrangements are there?
We solve the problem in the same way as before:
5
×
4
×
3
×
2
×
1
=
120
—–
—–
—–
—–
—–
1st
2nd
3rd
4th
5th
There are 120 different arrangements.
•
Notice that when we wished to arrange 6 people that it could be done
6
×
5
×
4
×
3
×
2
×
1 ways, and that when we wished to arrange 5
people, it could be done 5
×
4
×
3
×
2
×
1 ways.
•
In general, arranging
n
objects, can be done
n
×
(
n

1)
× · · · ×
2
×
1
ways.
•
These products have a special name, and a symbol.
1
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Suppose there are
n
distinct objects we wish to arrange.
The number of different arrangements is
n
!, which we read
as “
n factorial
”.
The symbol
n
! means the following:
n
! =
n
×
(
n

1)
× · · · ×
2
×
1.
n
! is defined for positive integer values of
n
. We also
define 0! = 1.
•
This tool helps us tackle certain counting problems quickly.
•
You should note that your calculator contains a factorial button, to
help speed up the solution of problems.
Example:
How many different ways are there to arrange the letters in
the word “OBJECTS”?
Since there are seven distinct letters in the word, then there are 7! = 5040
different arrangements.
Example:
Ten people get in line for a movie. How many different ways
are there for them to get in line?
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 Spring '11
 stephenlang
 Calculus, Permutations, Permutations And Combinations, access code, ways, Prize

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