A 1 ∪A 2 ∪…∪A n |=|A 1 |+|A 2 |+…+|A n
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Transcript A 1 ∪A 2 ∪…∪A n |=|A 1 |+|A 2 |+…+|A n
4.2 Permutations of sets
4.2.1
Basic counting principles
4.3(Addition principle): If A1, A2, … ,
An are disjoint sets, then the number of
elements in the union of these sets is the sum
of the numbers of elements in them.
| A1∪A2∪…∪An |=|A1|+|A2|+…+|An|
Theorem
Example1:
A student wishes to take either a
mathematics course or biology course, but not
both. If there are 4 mathematics course and 2
biology course for which the student has the
necessary prerequisites, then the student can
choose a course to take in 4+2=6 ways.
Theorem 4.4(Multiplication principle): Let A
and B be two finite sets. Let |A|=p and |B|=q,
then |A×B|=p×q.
Example2: A student
wishes to take a
mathematics course and a biology course. If
there are 4 mathematics course and 2 biology
course for which the student has the
necessary prerequisites, then the student can
choose courses to take in 4×2=8 ways.
4.2.2 Permutations of sets
An
ordered arrangement of r elements of an
n-element set is called an r-permutation.
We denote by p(n,r) the number of rpermutations of an n-element set. If r>n, then
p(n,r)=0. An n-permutation of an n-element
set S is called a permutation of S. A
permutation of a set S is a listing of the
elements of S in some order.
Theorem 4.5: For n and r positive integers with rn,
p(n,r)=n(n-1)…(n-r+1)
Proof:In constructing an r-permutation of an nelement set, we can choose the first item in n ways,
the second item in n-1 ways whatever choice of the
first item,… , and the rth item in n-(r-1) ways
whatever choice of the first r-1 items. By the
multiplication principle the r items can be chosen in
n(n-1)…(n-r+1) ways.
We define n! by
n!= n(n-1)…2•1
with the convention that 0!=1.Thus p(n,r)=n!/(n-r)!.
Example:What is the number of ways to order the
26 letters of the alphabet so that no two of the
vowels a,e,i,o,and u occur consecutively?(元音字母
中任意两个都不得相继出现)
Solution:The first task is to decide how to order the
consonants among themselves.
21!
Our second task is put the vowels in these places.
p(22,5)=22!/17!
By the multiplication principle, the numble of
ordered arrangements of the letters of the alphabet
with no two vowels consecutive is 21!22!/17! .
Example:
What is the number of ways to
order the 26 letters of the alphabet so that it
contains exactly seven letters between a and b?
a…….b,P(24,7)
b…….a,P(24,7)
between a and b
2P(24,7)
P(18,18)=18!.
2P(24,7)18!
linar permutation
circular permutation
linar permutation
circular permutation
linar
permutation 12345
linar permutation 45123
circular permutation
For example, the circular permutation
arises
from each of the linear
permutation
12345 23451 34512 45123 51234
Theorem 4.6: The number of circular rpermutations of a set of n elements is given
by p(n,r)/r=n!/r(n-r)! . In particular,the
number of circular permutations of n
elements is (n-1)! .
Proof: The set of linear r-permutations can
be partitioned into parts in such a way that
two linear r-permutations are in the same
part if only if they correspond to the same
circular r-permutations .
Thus the number of circular r-permutations
equals the number of parts.
Since each part contains r linear rpermutations, the number of parts is the
number p(n,r) of linear r-permutations
divided by r.
Example:
Ten people, including two who do
not wish to sit next to one another, are to be
seated at a round table. How many circular
seating arrangements are there?
4.3 Combinations of sets
4.3.1
Combinations of sets
Definition:
Let r be a non-negative integer. An
r-combination of a set S is an r-element subset
of S. We denote by nCr, or C(n,r), or
n
r
Example:
Let S={a,b,c,d}, then {a,b,c}, {a,b,d},
{a,c,d}, {b,c,d} are the 3-combinations of S.
Note that these are subsets, not sequences.
Therefore,
{a,b,c}={a,c,b}={b,a,c}={b,c,a}={c,a,b}
={c,b,a}.
If r>n, then C(n,r)=0. Also C(0,r)=0 if r>0.
Obviously
C(n,0)=1, C(n,1)=n, C(n,n)=1.
Theorem 4.7: For 0rn, C(n,r)=p(n,r)/r! and hence
C(n,r)=n!/r!(n-r)!.
Proof: Let S be an n-element set.
Each r-permutation of S arises in exactly one way as
a result of carrying out the following two tasks.
(1)Choose r elements from S. C(n,r)
(2)Arrange the chosen r elements in some order. r!
By the multiplication principle we have
p(n,r)=r!C(n,r). Thus C(n,r)=p(n,r)/r!. We now use
the formula p(n,r)=n!/(n-r)! and obtain
C(n,r)=n!/r!(n-r)!.
Corollary 4.1: For 0rn, C(n,r)=C(n,n-r).
Proof: C(n,r)=n!/(r!(n-r)!)=n!/((n-(n-r))!(nr)!)=C(n,n-r).
Example: How many different seven-person
committees can be formed each containing three
women from an available set of 20 women and four
men from an available set of 30 men?
Task1: Choose three women from the set of 20
women. C(20,3)
Task2: Choose four men from an the set of 30 men.
C(30,4)
By the multiplication principle,there are
C(20,3)C(30,4).
4.3.2 The
Binomial Coefficients and Identities
The number C(n,r) have many important and
fascinating properties.
C(n,r) is also called a binomial coefficient
because these numbers occur as coefficients in
the expansion of powers of binomial
expressions such as (a+b)n.
Theorem 4.8(Binomial theorem): Let x and y
be variables, and let n be a nonnegative
n
integer. Then
( x y ) n C ( n, k ) x k y n k
k 0
Corollary
4.2: Let n be a nonnegative integer.
Then
C(n,0)+C(n,1)+…+C(n,n)=2n
Proof:Let x=y=1, by the Binomial theorem it
follows that
Corollary
4.3:Let n be a positive integer. Then
C(n,0)-C(n,1)+C(n,2)-…+ (-1)nC(n,n)=0
Proof:Let x=-1, and y=1, by the Binomial
theorem it follows that C(n,0)-C(n,1)+C(n,2)…+ (-1)nC(n,n)=0
Remark: Corollary 4.3 implies that
C(n,0)+C(n,2)+ …=C(n,1)+C(n,3)+ …
Theorem 4.9: Let m,n,r, and k be nonnegative integer. Then
(1)C (n, k )
n
C (n 1, k 1)
k
(2)C(n,k)=C(n-1,k)+C(n-1,k-1)(杨辉公式,Pascal’s formula);
n
(3) kC (n, k )
n 2n 1
k 1
n
(4) k 2C (n, k ) n(n 1)2n 2
k 1
(5)C(n,r)C(r,k)=C(n,k)C(n-k,r-k), where rk;
(6)C(m,0)C(n,r)+C(m,1)C(n,r-1)++C(m,r)C(n,0)=C(m+n,r),
where rmin{m,n} (Vandermonde identity);
(7)C(m,0)C(n,0)+C(m,1)C(n,1)++C(m,m)C(n,m)=C(m+n,m)where
n
mn。When m=n,
2
C ( n, k )
k 0
m
(8) C (n k , k ) C (n m 1, m)
k 0
C ( 2 n, n )
4.4 Permutations and Combinations of multisets
Multisets :A multiset is a set in which an item may
appear more than once. .
Sets and multisets are quite similar: both support
the basic set operations of union, intersection, and
difference.
item ai
ni,
{n1•a1,n2•a2,…,nk•ak}
Example:{a,a,a,a,b,b,c}
{4•a,2•b,1•c}
{•a1,•a2,…,•ak}
4.4.1
If
Permutations of multisets
S is a multiset, a r-permutation of S is an
ordered arrangement of r of the objects of S.
If the total number of objects of S is n,then a
n-permutation of S will also be called a
permutation of S.
For example, if S={2•a,1•b,3•c}, then
aacb acbc cacc are 4-permutations of S,
“abccac” is a permutation of S.
The multiset S has no 7-permutations since
7>2+1+3=6, the number of objects of S.
Theorem 4.10: Let S be a multiset with objects of k
different types where each has an infinite repetition
number. Then the number of r-permutations of S is
kr.
Proof: In constructing a r-permutation of S,
we can choose the first item can be an object of any
one of the k types.
Similarly the second item to be an object of any one
of the k types, and so on.
Since all repetition numbers of S are infinite, the
number of different choices for any item is always k
and does not depend on the choices of any previous
items.
By the multiplication principle, the r items can be
chose in kr ways.
4.4: Let S={n1•a1,n2•a2,…,nk•ak},
and ni r for each i=1,2,…,n,then the
number of r-permutations of S is kr.
Corollary
Theorem
4.11: Let multiset
S={n1•a1,n2•a2,…,nk•ak}, and
n=n1+n2+…+nk=|S|. Then the number of
permutations of S equals n!/(n1!n2!…nk!)。
Proof: We can think of it this way. There are n
places, and we want to put exactly one of the
objects of S in each of the places.
Since there are n1 a1’s in S, we must choose a
subset of n1 places from the set of n places.
C(n,n1)
We next decided which places are to be
occupied by the a2’.
Example
What is the number of
permutations of the letters in the word
Mississippi?
S={n1•a1,n2•a2,…,nk•ak}, and
n=n1+n2+…+nk=|S|,then the number N of rpermutations of S equals
(1)0
r>n
(2)n!/(n1!n2!…nk!)
r=n
(3)kr
ni r for each i=1,2,…,n
(4)If r<n, there is, in general, no simple
formula for the number of r-permutations of
S.
Nonetheless a solution can be obtained by the
technique of generating functions, and we
discuss this in 4.6 .
Let
Combinations
of multisets P85-86 3.2
Inclusion-Exclusion principle and
Applications
Exercise P83
17, 19,22,33; P86 4,5,6,8
1.In how many ways can six men and six ladies be
seated round table if the men and ladies are to sit
in alternate seats?
2.In how many ways can 15 people be seated at a
round table if B refuse to sit next to A?What if B
only refuses to sit on A’s right?
3.Prove Theorem 4.9 (4)(7).
4. How many ways are there to assign three jobs to
five employees if each employee can be given more
than one job?
5.A book publisher has 3000 copies of a discrete
mathematics book . How many ways are there to store
these books in there three warehouses if the copies of
the book are indistinguishable?