5.3 Permutations and Combinations
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Urn models
We are given set of n objects in an urn (don’t ask
why it’s called an “urn” - probably due to some
statistician years ago) .
We are going to pick (select) r objects from the urn in
sequence. After we choose an object
-- we can replace it-(selection with replacement)
-- or not -(selection without replacement).
If we choose r objects, how many different possible
sequences of r objects are there?
Does the order of the objects matter or not?
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Permutations
• Selection without replacement of r objects from the
urn with n objects.
• A permutation is an arrangement.
Order matters .
• The number of permutations of n things taken r at a
time
P(n,r) = n(n - 1)(n - 2) . . . (n - r + 1)
• Note:
P(n, r) = n! /(n - r)!
• Example:
Let A and B be finite sets and let | A || B | .
Count the number of injections from A to B.
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Permutations
• Theorem 1: If n is a positive integer and r is an
integer with 1  r  n, then there are
P(n , r) = n(n-1)(n-2) ‥ (n-r+1)
r- permutations of a set with n distinct elements.
• Corollary 1: If n and r are integers with 0  r  n ,
then P(n, r)= n! / (n-r)!
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Permutations
• A permutation of a set of distinct objects is an ordered
arrangement of these objects.
• We also are interested in ordered arrangements of some
of the elements of a set.
• An ordered arrangement of r elements of a set is called
an r-permutation.
• Example 2: let S={1, 2, 3}. The ordered arrangement 3, 1,
2 is a permutation of S. The ordered arrangement 3, 2 is
a 2-permutation of S.
• Example 4: How many ways are there to select a firstprize winner, a second-prize winner, and a third-prize
winner from 100 different people who have entered a
contest?
• HW: example 7,(p.357)
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Combinations
• Selection is without replacement but
order does not matter
• It is equivalent to selecting subsets of size r from a set of
size n.
• Divide out the number of arrangements or permutations
of r objects from the set of permutations of n objects
taken r at a time:
• The number of combinations of n things taken r at a
 n  P(n, r )
n!
time
C (n, r ) =   =
=
r
 
Other names for C(n, r):
• n choose r
• The binomial coefficient
P(r , r )
r!(n - r )!
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Combinations
• Theorem 2: The number of r-combinations of a set
with n elements, where n is a nonnegative integer
and r is an integer with 0  r  n, equals
C(n, r) = n! / r! (n-r)!
• Corollary 2: Let n and r be nonnegative integers with
r  n. Then C(n, r) = C(n, n-r).
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Combinations
• Example:
How many subsets of size r can be constructed from a
set of n objects?
• Corollary: r=0n C(n, r)=2n
• Proof:
• If we count the number of subsets of a set of size n,
we get the cardinality of the power set.
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Combinations
• Example 11: How many poker hands of five cards can
be dealt from a standard deck of 52 card? Also, how
many ways are there to select 47 cards from a
standard deck of 52 cards?
• Example 13: A group of 30 people have been trained
as astronauts to go on the first mission to Mars. How
many ways are there to select a crew of six people to
go on this mission ( assuming that all crew members
have the same job)?
• Example 14: How many bit strings of length n contain
exactly r 1s?
• HW: Example 15, p(360)
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