§5.1 The Basics of counting

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Discrete

Mathematics

Chapter 5 Counting

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§5.1 The Basics of counting

Ú We will present two basic counting principles, the product rule and the sum rule.

Ú The product rule:

Suppose that a procedure can be broken down into two tasks. If there are n1 ways to do the

first task and n2 ways to do the second task after the first task has been done, then there are

n1 n2 ways to do the procedure.

n₁

n₂

n₁× n₂ ways

Example 1 A new company with just two employees, Sanchez and Patel, rents a floor of a building with 12 offices. How many ways are there to assign different offices to these two employees?

Sol: 12*11 =132 ways

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Example 2 The chair of an auditorium (大禮堂) is to be labeled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labeled differently?

Sol: 26 × 100 = 2600 ways to label chairs.

letter

Ν

∈

≤

≤ x

x 100 1

Example 4 How many different bit strings are there of length seven?

Sol: 1 2 3 4 5 6 7

□ □ □ □ □ □ □

↑ ↑ ↑ ↑ ↑ ↑ ↑ 0,1 0,1 0,1 . . . 0,1

→ 2⁷ 種

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Example 5

How many different license plates (車牌) are

available if each plate contains a sequence of 3 letters followed by 3 digits ?

Sol: □ □ □ □ □ □ →26³．10³ letter digit

Example 6 How many functions are there from a set with m elements to one with n elements?

Sol: f(a₁)=? 可以是b₁～ b_n, 共n種 f(a₂)=? 可以是b₁～ b_n, 共n種

：

f(a_m)=? 可以是b₁～ b_n, 共n種

∴n^m

a₁ a₂

．

a_m

b₁ b₂

．

b_n f

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Example 7 How many one-to-one functions are there from a set with m elements to one with n element?

(m ≤ n)

Sol: f(a₁) = ? 可以是b₁～ b_n, 共 n 種

f(a₂) = ? 可以是b₁～ b_n, 但不能= f(a₁), 共 n−1 種

f(a₃) = ? 可以是b₁～ b_n, 但不能= f(a₁), 也不能=f(a₂), 共 n−^{2 種}

：

f(a_m) = ? 不可=f(a₁), f(a₂), ... , f(a_m−1), 故共n−(m−1)種

∴共 n．(n−1)．(n−2)．...．(n−m+1)種 1-1 function _#

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Ú

A password on a computer system consists of 6,7, or 8 characters. Each of these characters must be a digit or a letter of the alphabet. Each password must contain at least one digit. How many such passwords are there?

Ú This section introduces

-

a variety of other counting problems

-

the basic techniques of counting.

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Basic Counting Principles

Ú The sum rule:

If a first task can be done in n₁ ways and a second task in n₂ ways, and if these tasks cannot be done at the same time. then there are n₁+n₂ ways to do either task.

Example 11

Suppose that either a member of faculty or a student is chosen as a representative to a university committee. How many different

choices are there for this representative if there are 37 members of the faculty and 83 students?

Sol: 37+83=120

n₁ n₂

n₁+ n₂ways

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Example 12 A student can choose a computer

project from one of three lists. The three lists contain 23, 15 and 19 possible projects respectively.

How many possible projects are there to choose from?

Sol: 23+15+19=57 projects.

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Example 14 Each user on a computer system has a password which is 6 to 8 characters long, where each character is an uppercase letter or a digit. Each

password must contain at least one digit.

How many possible passwords are there?

Sol: P_i: # of possible passwords of length i , i=6,7,8 P₆ = 36⁶− 26⁶

P₇ = 36⁷− 26⁷

P₈ = 36⁸− 268

∴ P₆ + P₇ + P₈ = 36⁶+ 36⁷+ 36⁸− 26⁶ − 26⁷− 26⁸種

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Example 13

In a version of Basic, the name of a variable

is a string of one (only alphanumeric characters )or two alphanumeric characters, (An alphanumeric characters is either one of the 26 English letters or one of the 10 digits) where uppercase and lowercase letters are not distinguished. Moreover, a variable name must begin with a letter and must be different from the five strings of two characters that are reserved for programming use.

How many different variable names are there in this version of Basic?

Sol:

Let V_i be the number of variable names of length i.

V₁=26

V₂=26．36 – 5

∴26 + 26．36 – 5 different names.

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※ The Inclusion-Exclusion Principle (排容原理)

A B

Example 17 How many bit strings of length eight either start with a 1 bit or end with the two bits 00 ?

Sol:

１２３４５６７８

□ □ □ □ □ □ □ □

↑ ↑ . . .

① 1 0,1 0,1 → 共2⁷種

② . . . 0 0 → 共2⁶種

③ 1 . . . 0 0 → 共2⁵種

∴ 2⁷+2⁶−2⁵種

B A

B

A U = + − I

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0

1

bit 1

※ Tree Diagrams

Example 19 How many bit strings of length four do not have two consecutive 1s ?

Sol:

Exercise: 11, 17, 21, 27, 36, 37, 45, 49

0 0 0 0 0 1 1 0 1

0 1

(0000) (0001) (0010) (0100) (0101) (1000) (1001) (1010)

∴ 8 bit strings 0

0 1 1

0

bit 3

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Ex 36. How many subsets of a set with 100 elements have more than one element ?

Sol:

Ex 37. A palindrome (迴文) is a string whose reversal is identical to the string. How many bit strings of length n are palindromes ?

(e.g., abcdcba 是迴文, abcd 不是 ) Sol: If a₁a₂ ... a_n is a palindrome, then

a₁=a_n, a₂=a_n−1, a₃=a_n−2, …

101 2

2

... 100 98

100 99

100 100

100 )

1

( ⎟⎟ = = ¹⁰⁰ −

⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟ +

⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝ + ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ L

Thm. 4 of §5.3

{

₁^, ₂ ^,..., ₁₀₀

}

) 2

( a a a

{

^□^,^□^,...,^□

}

² ¹⁰¹

:

subset ∴ ¹⁰⁰ −

放不放

空集合及

只有1個元素的集合

string.

種 2 ^{2 ⎥}^⎥

⎢⎢ ⎤

⎡

⇒

n

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§5.2 The Pigeonhole Principle (鴿籠原理)

Thm 1 (The Pigeonhole Principle)

If k+1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.

Proof

Suppose that none of the k boxes contains more than one object. Then the total number of objects would

be at most k. This is a contradiction.

Example 1. Among any 367 people, there must be at least two with the same birthday, because there are only 366 possible birthdays.

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Example 2 In any group of 27 English words, there must be at least two that begin with the same letter.

Example 3 How many students must be in a class to guarantee that at least two students receive the

same score on the final exam ? (0~100 points) Sol: 102. (101+1)

Thm 2. (The generalized pigeon hole principle) If N objects are placed into k boxes, then

there is at least one box containing at least objects.

e.g. 21 objects, 10 boxes ⇒ there must be one box

containing at least objects.

⎥⎥⎤

⎢⎢⎡ k N

1021 =3

⎥⎥⎤

⎢⎢⎡

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Example 5 Among 100 people there are at least

who were born in the same month. ( 100 objects, 12 boxes )

12 9 100 =⎥⎥⎤

⎢⎢⎡

Example 6 What is the minimum number of students required in a D.M class to be sure that at least six will

receive the same grade, if there are five possible grades, A, B, C, D and E?

Sol: [N/5] = 6; N= 5*5+1 =26

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Example 10 During a month with 30 days a baseball team plays at least 1 game a day, but no more

than 45 games. Show that there must be a period

of some number of consecutive days during which the team must play exactly 14 games.

day 1 2 3 4 5 ... 15 30

# of game 3 2 1 2

sum ≤ 45

存在一段時間的game數和=14

Some Elegant Applications of the Pigeonhole Principle

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Sol:

Let a_j be the number of games played on or before the jth day of the month. (第1天～第j天的比賽數和)

Then is an increasing sequence of distinct integers with

Moreover, is also an increasing sequence of distinct integers with

There are 60 positive integers

between 1 and 59. Hence, such that

) 30 1

( ≤ j ≤

30 2

1

, a ,..., a a

j a_j ≤ ∀

≤ 45 1

) 45 ...

1 ., .

(i e ≤ a₁ < a₂ < a₃ < < a₃₀ ≤

14 ,...,

14 ,

14 ₂ ₃₀

1 + a + a +

a

59 14

15 ≤ a_j + ≤

) 59 14

...

14 14

14 15

., .

(i e ≤ a₁ + < a₂ + < a₃ + < < a₃₀ + ≤

14 ,...,

14 ,

,..., ₃₀ ₁ ₃₀

1 a a + a +

a j i and

∃

場) 共14

天天～第第 1

., . (

14 i e j i

a

a_i = _j + +

#

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Def. Suppose that is a sequence of numbers.

A subsequence of this sequence is a sequence of the form where

e.g. sequence: 8, 11, 9, 1, 4, 6, 12, 10, 5, 7 subsequence: 8, 9, 12 (9)

9, 11, 4, 6 (8)

Def. A sequence is called increasing (遞增) if A sequence is called decreasing (遞減) if

A sequence is called strictly increasing (嚴格遞增) if A sequence is called strictly decreasing (嚴格遞減) if

aN

a

a₁, ₂,...,

im

i

a a

a , ,...,

2 1

N i

i

i < < < _m ≤

≤ ...

1 ₁ ₂

)

., .

(i e 保持原順序

+1

≤ _i

i a

a

+1

≥ _i

i a

a

+1

< _i

i a

a

+1

> _i

i a

a

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Thm 3. Every sequence of n²+1 distinct real numbers contains a subsequence of length n+1 that is either strictly increasing or strictly decreasing.

Example 12. The sequence 8, 11, 9, 1, 4, 6, 12, 10, 5, 7 contains 10=3²+1 terms (i.e., n=3).

There is a strictly increasing subsequence of length four, namely, 1, 4, 5, 7. There is also a decreasing subsequence of length 4, namely, 11, 9, 6, 5.

Exercise 21 Construct a sequence of 16 positive integers that has no increasing or decreasing

subsequence of 5 terms.

Sol:

Exercise: 5, 13 (參考習題：21, 23)

13 , 14 , 15 , 16 9

, 10 , 11 , 12 5

, 6 , 7 , 8 1 , 2 , 3 , 4

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§4.3 Permutations(排列) and Combinations(組合)

Def. A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement of r elements of a set is called an r-permutation.

Example 1. Let The arrangement 3,1,2 is a permutation of S. The arrangement 3,2 is a

2-permutation of S.

Thm 1. The number of r-permutation of a set with n distinct elements is

位置： 1 2 3 … r

□ □ □ … □

↑ ↑ ↑ … ↑ 放法：

}.

3 , 2 , 1

= { S

)!

( ) ! 1 )...(

2 (

) 1 (

) ,

( n r

r n n n

n n r

n

P = ⋅ − ⋅ − − + = −

n

n−1

n − 2

ⁿ^−r ⁺¹

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Example 2’. How many different ways are there to select 4 different players from 10 players on a team to play

four tennis matches, where the matches are ordered ? Sol:

Example 4. Suppose that a saleswoman has to visit

8 different cities. She must begin her trip in a specified city, but she can visit the other cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities ?

Sol:

. 5040 7

8 9 10 )

4 , 10

( = × × × =

P

5040

!

7 =

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Def. An r-combination of elements of a set is an unordered selection of r elements from the set.

Example 6 Let S be the set {1, 2, 3, 4}.

Then {1, 3, 4} is a 3-combination from S.

Thm 2 The number of r-combinations of a set with n elements, where n is a positive integer and r is an integer with , equals

pf :

)!

(

!

! ) ,

)

(

( )

,

( n r

ⁿ_r ^p _rⁿ ^r _r _nⁿ _r

C

_rⁿ

= = = =

₋

n r ≤

≤ 0

! )

, ( )

,

( n r C n r r

P = ×

稱為 binomial coefficient

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Example 7. We see that C(4,2)=6, since the 2-combinations of {a,b,c,d} are the six subsets {a,b}, {a,c}, {a,d}, {b,c}, {b,d} and {c,d}

Corollary 1. Let n and r be nonnegative integers with r ≤ n.

Then C(n,r) = C(n,n−r) pf : From Thm 2.

組合意義：選 r 個拿走,相當於是選 n − r 個留下.

) ,

))! ( (

( )!

(

! )!

(

! ) !

,

( C n n r

r n n

r n

n r

n r r n

n

C = −

−

= −

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Example 8. How many ways are there to select 5

players from a 10-member tennis team to make a trip to a match at another school ?

Sol: C(10,5)=252

Example 11. How many ways are there to select a committee if the committee is to consist of 3

faculty members from the math department and 4 from the computer science department, if there are 9 faculty members of the math department and

11 of the computer science department ? Sol:

Exercise: 3, 11, 13, 21, 33, 34.

) 4 , 11 ( )

3 , 9

( C

C ×

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§4.5 Binomial Coefficients (二項式係數)

Example 1.

要產生 xy² 項時，

需從三個括號中選兩個括號提供 y，剩下一個則提供 x (注意：同一個括號中的 x 跟 y 不可能相乘)

∴共有 種不同來源的 xy²

⇒ xy² 的係數 =

∴

Thm 1. (The Binomial Theorem, 二項式定理)

Let x,y be variables, and let n be a positive integer, Then

3 2

2 3

3 ( )( )( ) ? ? ?

)

(x + y = x + y x + y x + y = x + x y + xy + y

) (³₂

3 3

3 2

3 2 2

3 1 3

3 0

3

( ) ( ) ( ) ( )

)

( x + y = x + x y + xy + y

∑

=

−

− + + + =

+

=

+ ⁿ

j

j j n n

j n

n n n

n n

n

n x x y xy y x y

y x

0 1

1 1

1

0) ( ) ... ( ) ( ) ( )

( )

(

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Example 4. What is the coefficient of x¹²y¹³ in the expansion of ?

Sol:

∴

Cor 1. Let n be a positive integer. Then

pf : By Thm 1, let x = y = 1

Cor 2. Let n be a positive integer. Then pf : by Thm 1. (1−1)ⁿ= 0

)25

3 2

( x − y

25

25 (2 ( 3 ))

) 3 2

( x − y = x + − y

13 12

25

13 ) 2 ( 3)

( ⋅ ⋅ −

) ( ...

) ( ) ( ) ( )

1 1

( +

ⁿ

=

₀ⁿ

+

₁ⁿ

+

₂ⁿ

+ +

ⁿ_n

∑

=

n −

k

n k k 0

0 )

( ) 1 (

∑

=

= +

+ +

n =

k

n n

k 0

1

0) ( ) ... ( ) 2 (

) (

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Thm 2. (Pascal’s identity)

Let n and k be positive integers with n ≥ k Then

⎟⎟⎠

⎜⎜ ⎞

⎝ +⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

= −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

k n k

n k

n

1 1

PASCAL’s triangle

) (⁰₀

)

(¹₀ (¹₁)

)

( ₀² (₁² ) (²₂ )

)

(³₀ (₁³ ) (³₂ ) (³₃ )

) (₃⁴

…

1

1 1

1 2 1

1 3 3 1

… 4

1 4 6 1

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pf : ①利用來證

②(組合意義):

)!

(

! ) !

,

( j i j

j i i

C = −

Suppose that T is a set containing n+1 elements.

Let and

A subset of T with k elements either contains a together with k−1 elements of S,

or contains elements of S. _# T

a∈ S = T −{a}.

n

‧ a

k

取法

=

a

k n

‧ 0

a

k−1 +

n

‧ 1

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Thm 3. (Vandermode’s Identity)

pf :

∑

= +

⋅

−

= +

≤

≤ Ζ

∈

r

k

k n C k

r m C r

n m

C

n m r

r n m

0

) , ( )

, ( )

, (

, 0

, ,

,

∑

=

⋅

−

=

= +

r

k

k n C k

r m C r

n m

C

0

) , ( )

, (

個的方法取

) (

m n

m n m n m n

↓↓ + ↓↓ +...+ ↓↓

0, r 1, r−1 r, 0

)!

(

!

! ) ,

) (

( ) ,

(n r ⁿ_r ^p _rⁿ ^r _r _nⁿ _r C

C_rⁿ = = = = ₋

(31)

Ex 33. Here we will count the number of paths between the origin (0,0) and point (m,n) such that each path is made up of a series of

steps, where each step is a move one unit to the right or a more one unit upward.

1 4 10 20 35 56 (5,3) 1 3 6 10 15 21

1 2 3 4 5 6 (0,0) 1 1 1 1 1

0 1 1 0 0 0 1 0

Each path can be Represented by a bit String consisting of m Os and n 1s.

(→) (↑) 56 8 7

! 3

! 5

! 8 5

3 5

=

×

=

⎟ =

⎠⎞

⎜⎝

⎛ +

There are paths of the desired type._⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + n

n m

Exercise: 7, 21

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§5.5 Generalized Permutations and Combinations

∴⎜⎝⎛ + − ⎟⎠⎞ 種方法 _# r

n

r 1

Thm 1. The number of r-permutations of a set of n objects with repetition allowed is n^r.

Thm 2. There are r-combinations from a set with n elements when repetition of elements is allowed.

pf : (視為有 r 個＊，要放入 n 個位置，即需插入 n−1 個bar 將之隔開)

) , 1 (r n r C + −

＊＊｜＊｜｜＊＊＊

a₁出現2次

a₂出現1次 a3不出現

a₄出現3次

} ,

, , , ,

{a₁ a₁ a₂ a₄ a₄ a₄

取法

} ,

, ,

{a₁ a₂ a₃ a₄ 例：設n = 4, 集合為

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Example 4. Suppose that a cookie shop has 4 different kinds of cookies. How many different ways can

6 cookies be chosen?

Sol: 6個cookie插入3個bar ⇒ 種

Example 5. How many solutions does the equation have, where are nonnegative integers?’

Sol: 11個1間要插入2個bar

⎟⎠

⎜ ⎞

⎝⎛ + 6

3 6

3 11

2

1 + x + x = x

3 2 1, x , x x

⎟⎠

⎜ ⎞

⎝⎛ + 11

2

⇒ 11 種

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※若上題改要求則原式

可改成

此時相當於是

其中且

∴5個1間要插入2個bar 種 (注意： case

相當於 )

※若上題再改為

則需排除 (即的情況)

因

∴共有種

3

, 2

,

1 ₂ ₃

1 ≥ x ≥ x ≥

x

3 11

2

1 + x + x = x

5 3 2 1 11 )

3 (

) 2 (

) 1

(x₁ − + x₂ − + x₃ − = − − − =

3 5

2

1 + y + y =

y y₁ = x₁ −1, y₂ = x₂ −2, … Ν

3 ∈

2 1, y , y y

⎟⎠

⎜ ⎞

⎝⎛ + 5

2 5

1 y , 3

,

1 ₂ ₃

1 = y = =

y

4 x

, 5

,

2 ₂ ₃

1 = x = =

x

3 x

, 2

, 3

1≤ x₁ ≤ x₂ ≥ ₃ ≥

1 > 3

x x₁ > 4

2 9 11 )

3 (

) 2 (

) 4

(x₁ − + x₂ − + x₃ − = − =

⎟⎠

⎜ ⎞

⎝⎛ +

⎟−

⎠⎞

⎜⎝

⎛ +

2 2 2 5

2 5

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※ Permutations with indistinguishable objects

Example 8. How many different strings can be made by reordering the letters of the word

SUCCESS ? Sol:

有3個S, 2個C, 1個U 及 1個E,

可放S的位置有種

剩下的4個位置中可放C的有種

剩下的2個位置中可放U的有種

剩下的1個位置中可放E的有種

∴共種

) (⁷₃

) (⁴₂

) (₁²

) (¹₁

) )(

)(

(⁷₃ ⁴₂ ₁² ¹₁

(36)

Thm 3. The number of different permutations of n objects, where type 1： n₁種

type 2： n₂種 is type k： n_k種

pf :

※ Distributing objects into Boxes

Example 9. How many ways are there to distribute hands of 5 cards to each of four players from the standard deck of 52 cards?

Sol: player 1：種

player 2：從剩下的牌再發5張

!

2

1 n nk

n

M L

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − − − −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎟⎟⎛ −

⎠

⎜⎜ ⎞

⎝

⎛ ₋

k

n

n n

n n n

n n n n

n n n

n ₁ ₂ ₁

3 2 1 2

1 1

　　L L

!

2

1 n k

n n

　L

=

⎟⎠

⎜ ⎞

⎝⎛ 5 47

⎟⎠

⎜ ⎞

⎝⎛ 5 52

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注意：上題相當於將52張牌放進5個不同的box的放法, 即 box 1 的給 player 1

box 2 的給 player 2 box 3 的給 player 3 box 4 的給 player 4 而 box 5 的是剩下的牌.

Thm 4. The number of ways to distribute n distinguishable objects into k distinguishable boxes so that n_i

objects are placed into box equals (跟Thm 3相等)

Exercise: 15, 20, 17, 25, 31, 55

!

2

1 n nk

n

n L

k i

i, =1,2,L,

! 32

! 5

! 52

! 32

! 5

! 37

! 5

! 42

! 5

! 47

! 5

! 52 5

37 5

42 5

47 5

52⎟⎠⎞⎜⎝⎛ ⎟⎠⎞⎜⎝⎛ ⎟⎠⎞⎜⎝⎛ ⎟⎠⎞ = ⋅ ⋅ ⋅ =

⎜⎝

∴⎛