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PermutationsandCombinations_VFIN.doc
Permutations and Combinations or the Mathematics of
Hugs and Kisses
1. An apology
This is a topic that I use to discuss much earlier in the course, because I had the
impression that it was very important in leading up to the discussion of probabilistic
models and language models which we have already talked about. Eventually, I learned
that those could be discussed without having talked about permutations and
combinations at all, and this very interesting topic almost fell completely out of the
course. However, I realized that it has some interesting applications in examining the
complexity of networks of relations among people, and among documents. So it
managed to fight its way back into the course, but a little bit later in the syllabus.
2. Permutations
Let’s begin by thinking about permutations. The number of permutations that can be
made with some number
n
of items means the number of different orders that we can
place them in.
2.1.
A simple example
For example if we have 3 items that are named A, B and C
We notice that while we’re still keeping A in the first position
A B C
We could switch the second two and get a new order like this.
A B C
But of course we could also have done that with B in the first position
B A C
And then:
B C A
And so we would get two more orders. And finally we could have done it with C in the
first position getting two additional orders.
C A B
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C BA
Altogether we see that there are 6 different orders that can be made with those three
different things. And it does not seem that there are any more. So we can write that the
number of permutations of 3 objects is given by the equation
6
n
P
.
2.2.
What happens if we add another object to the set?
Now we are going to want to apply some kind of mathematical induction here so we’d
like to ask the question: “if we already know a formula for
n
P
can we calculate the
number
1
n
P
?”
I will argue that we can.
If we have one more item that we need to put into the list, let’s call it X, we see that for
any one of the lists of orders that we have made, for example A C B, there are four
different ways that we could put the X into the list.
We could put it in front of any one of the letters; that makes 3 ways. Or we could put it
after all of them and that’s the fourth way.
That means that if we add a fourth item to the set of items, each of our old orders of the
3 items can be turned into 4 different ones.
And it’s clear that all 4 of those will be different from each other. This means that we
can write the equation
43
4
PP
.
Now, in fact, there’s nothing at all special about the fact that we went from 3 to 4; no
matter how many items we had in the list if we were going to add one more there would
be
1
n
places that we could put it “into” the list.
So we can actually write the equation that says that
1
(
1)
nn
P
n
P
.
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 Fall '09
 Boros

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