Permutations and Combinations

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Permutations and Combinations
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Many counting problems can be easily solved if we
can identify them as permutation or combination
problems.
Definition
A permutation of r objects selected from a set
of n objects is an arrangement of r of the
n objects in a specific order.
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The number of permutations of r objects
selected from a set of n objects will be
denoted by nPr.
The good news is that we can use our graphing
calculators to compute nPr.
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Example
How many ways are there to choose the first,
second, and third prize winners in a beauty
contest with 15 entrants?

• This problem can be done
• by using the multiplication principle
• by identifying it as a permutation problem

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Example
You have 15 compact discs. Your girlfriend wants
to borrow 3 of them. In how many different ways
can she select the 3 CDs?
Observe that there is no indication of order in
this problem. So it is not a permutation
problem. It is an example of a combination
problem.
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Definition
A combination of r objects selected from a set
of n objects is a selection of r of the n
objects with order disregarded.
The number of combinations of r objects
selected from a set of n objects will be
denoted by nCr.
Calculators can be used to compute nCr, just as
we've used them to compute nPr.
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Now we can easily solve the problem
Example
You have 15 compact discs. Your girlfriend wants
to borrow 3 of them. In how many different ways
can she select the 3 CDs?
You will find it useful to practice
distinguishing combination from permutation
problems. But remember that in permutation
problems, order is indicated in one way or
another.
Phrases like arrange, schedule, line up, and so
on, suggest the existence of some order.
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Example

• A 5-person committee is to be formed from a
  group of 10 female and 7 male executives.
• How many 5-person committees are possible?
• How many of those committees contain 3 females
  and 2 males?
• How many of those committees contain no male?
• How many of those committees contain at most 1
  male?