2. Combinatorial Methods

p2.

2.1 Introduction

• If the sample space is finite and furthermore sample points are all equally likely, then

P(A)=N(A)/N(S) So we study combinatorial analysis here, which deals with methods of counting.

p3.

2.2 Counting principle Ex 2.1 How many outcomes are there if we throw 5 dice?

Ex 2.2 In tossing 4 fair dice, P(at least one 3 among these 4 dice)=?

Ex 2.3 Virginia wants to give her son, Brian, 14 different baseball cards within a 7-day period. If Virginia gives Brian cards no more than once a day, in how many way can this be done?

Ex 2.6 (Standard Birthday Problem) P(at least two among n people have the same Bday)=?

p4.

Counting principle Thm 2.3 A set with n elements has 2n subsets.

Ex 2.9 Mark has $4. He decides to bet $1 on the flip of a fair coin 4 times. What is the probability that (a) he breaks even; (b) he wins money?(use tree diagram)

p5.

2.3 Permutations Ex 2.10 3 people, Brown, Smith, and Jones, must be

scheduled for job interviews. In how many different orders can this be done?

Ex 2.11 2 anthropology, 4 computer science, 3 statistics, 3 biology, and 5 music books are put on a bookshelf with a random arrangement. What is the probability that the books of the same subject are together?

p6.

Permutations Ex 2.12 If 5 boys and 5 girls sit in a row in a random order,

P(no two children of the same sex sit together)=?

Thm 2.4 The number of distinguishable permutations of n objects of k different types, where n1 are alike, n2 are alike, …, nk are alike and n=n1+n2+…+nk is

!!...!

!

21 knnn

n

p7.

Permutations Ex 2.13 How many different 10-letter codes can be made

using 3 a’s, 4 b’s, and 3 c’s?

Ex 2.14 In how many ways can we paint 11 offices so that 4 of them will be painted green, 3 yellow, 2 white, and the remaining 2 pink?

Ex 2.15 A fair coin is flipped 10 times. P(exactly 3 heads)=?

p8.

2.4 Combinations

Definition

An unordered arrangement of r objects from a set A containing n objects (r n) is called an r-element combination of A, or a combination of the elements of A taken r at a time.Notes ：

)!(!

!

rnr

n

r

nC n

r

1

1

!

r

n

r

n

r

n

CrP nr

nr

p9.

2.4 Combinations Ex 2.16 In how many ways can 2 math and 3 biology books

be selected from 8 math and 6 biology books?

Ex 2.17 45 instructors were selected randomly to ask whether they are happy with their teaching loads. The response of 32 were negative. If Drs. Smith, Brown, and Jones were among those questioned. P(all 3 gave negative responses)=?

p10.

Combinations Ex 2.18 In a small town, 11 of the 25 schoolteachers are

against abortion, 8 are for abortion, and the rest are indifferent. A random sample of 5 schoolteachers is selected for an interview. (a)P(all 5 are for abortion)=? (b)P(all 5 have the same opinion)=?

Ex 2.19 In Maryland’s lottery, player pick 6 integers between 1 and 49, order of selection being irrelevant.

P(grand prize)=? P(2nd prize)=? P(3rd prize)=?

p11.

Combinations Ex 2.20 7 cards are drawn from 52 without replacement.

P(at least one of the cards is a king)=?

p12.

E.g. 7 個人買食物 , 有四種食物可選擇 , 有幾種買法 ?

first second third fourth

xxx xxxx

xx x x xxx

xxxx xxx

7

174

!3!7!10

for x for

Combinations with Repetition: Distributions

p13.

In general, the number of selections, with repetitions, of r objects from n distinct objects are:

( )!

!( )!

n r

r n

n r

r

1

1

1

Combinations with Repetition: Distributions

p14.

E.g. Determine all integer solutions to the equation

x x x x1 2 3 4 7 , where xi 0 for all 1 4 i .

select with repetition from x x x x1 2 3 4, , , 7 times

For example, if x1 is selected twice, then x1 2

in the final solution. Therefore, C(4+7-1,7)=120

Combinations with Repetition: Distributions

p15.

Equivalence of the following:

(a) the number of integer solutions of the equationnixrxxx in 1 ,0 ,21

(b) the number of selections, with repetition, of size r from a collection of size n

(c) the number of ways r identical objects can be distributed among n distinct containers

Combinations with Repetition: Distributions

p16.

y y y y1 2 6 7 9 ,

E.g. How many nonnegative integer solutions are there tothe inequality x x x1 2 6 10 ?

77621 0 ,61 ,0 ,10 xixxxxx i

It is equivalent to

which can be transformed to

where y xi i for

1 6 i

and

y x7 7 1 C(7+9-1,9)=5005

Combinations with Repetition: Distributions

p17.

E.g. How many terms there are in the expansion of

( )w x y z 10?

Each distinct term is of the form

10

1 2 3 4

1 2 3 4

n n n nw x y zn n n n

, , ,,

where 0 n i for

1 4 i , and n n n n1 2 3 4 10 .

Therefore, C(4+10-1,10)=286

Combinations with Repetition: Distributions

p18.

Theorem 2.5 ： Binomial Expansion For any integer n 0,

Pf ：.)(

0

iinn

i

n yxi

nyx

p19.

Combinations Thm 2.5 (Binomial expansion)

Ex 2.25 What is the coefficient of x2y3 in the expansion of (2x+3y)5?

Ex 2.26 Evaluate the sum

iinn

i

n yxi

nyx

0

)(

.3210

n

nnnnn

p20.

Combinations Ex 2.27 Evaluate the sum

Ex 2.28 Prove that

.3

32

21

n

nn

nnn

.2

2

0

n

ii

n

n

n

p21.

Combinations Thm 2.6 (Multinomial expansion).

k

k

nk

nn

nnnn k

nk xxx

nnn

nxxx 21

21

21... 21

21 !!...!

!)(