# GMAT数学之光—排列组合的本质

## 基本概念

6*5*4*3*2*1

6*5*4*3*2*1

6*5*4*3*2*1 = 6!

6*5

2A6或2P6

（因为帖子里打不出真正的写法，我们就简单写为这样。打包版PDF中有正确的写法）

Each participant in a certain study was assigned a sequence of 3 different letters from the set {A, B, C, D, E, F, G, H}.  If no sequence was assigned to more than one participant and if 36 of the possible sequences were not assigned, what was the number of participants in the study?  (Note, for example, that the sequence A, B, C is different from the sequence C, B, A.)

(A) 20

(B) 92

(C) 300

(D) 372

(E) 476

## 除序法

6*5/2!

“2!”表示两个小球的全排列。6*5中计算了两个小球的顺序，而因为题目不考虑它们的顺序，所以将该顺序除掉。这种不计算取出的小球间的顺序的方式叫做“组合”，记作：

2C6

（因为帖子里打不出真正的写法，我们简写为这样。打包版PDF中有正确的写法）

“除”顺序而不是“减”顺序的原因是取出的两个元素的顺序只是整个任务的一个步骤，我们可以把它想象成先取出小球，之后再对取出的小球进行全排列。根据乘法原理则有：

Departments A, B, and C have 10 employees each, and department D has 20 employees.  Departments A, B, C, and D have no employees in common.  A task force is to be formed by selecting 1 employee from each of departments A, B, and C and 2 employees from department D.  How many different task forces are possible?

(A) 19,000

(B) 40,000

(C) 100,000

(D) 190,000

(E) 400,000

2C10 = 20*19/2! = 190

From a group of 21 astronauts that includes 12 people with previous experience in space flight, a 3-person crew is to be selected so that exactly 1 person in the crew has previous experience in space flight. How many different crews of this type are possible?

(A) 432

(B) 594

(C) 864

(D) 1,330

(E) 7,980

## 重复元素

There are 5 cars to be displayed in 5 parking spaces, with all the cars facing the same direction. Of the 5 cars, 3 are red, 1 is blue, and 1 is yellow. If the cars are identical except for color, how many different display arrangements of the 5 cars are possible?

(A) 20

(B) 25

(C) 40

(D) 60

(E) 125

## 桶装信问题

A certain company assigns employees to offices in such a way that some of the offices can be empty and more than one employee can be assigned to an office.  In how many ways can the company assign 3 employees to 2 different offices?

(A) 5

(B) 6

(C) 7

(D) 8

(E) 9

## 插空，捆绑法

AACDE五个字母排队，要求两个A必须不能挨着，问一共几种排法？

The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?

(A) 12

(B) 18

(C) 24

(D) 36

(E) 48

## 隔板法

0 0 0 0 0 0

0 I 0 0 0 I 0 0

If x, y, z are both positive integers and x + y + z = 10, how many different sets of x, y, z?

(A) 72

(B) 54

(C) 36

(D) 20

(E) 18

x, y, z均不能为0且是整数，一共加和为10，这就可以看作10个苹果分三组，每组必有一个苹果的情况。10个苹果间有9个空位，即，2C9 = 36。答案为C。

## 圆桌问题

At a dinner party, 5 people are to be seated around a circular table. Two seating arrangements are considered different only when the positions of the people are different relative to each other. What is the total number of different possible seating arrangements for the group?

(A) 5

(B) 10

(C) 24

(D) 32

(E) 120

## 实战解题

A company plans to assign identification numbers to its employees. Each number is to consist of four different digits from 0 to 9, inclusive, except that the first digit cannot be 0. How many different identification numbers are possible?

(A) 3,024

(B) 4,536

(C) 5,040

(D) 9,000

(E) 10,000

A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department.  If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?

(A) 42

(B) 70

(C) 140

(D) 165

(E) 315

1C7*2C10 = 315。答案为E。

Two members of a club are to be selected to represent the club at a national meeting. If there are 190 different possible selections of the 2 members, how many members does the club have?

(A) 20

(B) 27

(C) 40

(D) 57

(E) 95

The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

(A) 96

(B) 120

(C) 144

(D) 180

(E) 240

2C5*4！=240。答案为E。

In a certain group of 10 members, 4 members teach only French and the rest teach only Spanish or German. If the group is to choose a 3-member committee, which must have at least 1 member who teaches French, how many different committees can be chosen?

(A) 40

(B) 50

(C) 64

(D) 80

(E) 100

10人中有4人只教法语，剩下6人只教西语或者德语。从10人中选三人成小队，至少有一个人教法语。问有多少种选法？

1.仅有1个法语老师：1C4*2C6=60；

2.有且仅有2个法语老师：2C4*1C6=36；

3.3个都是法语老师：3C4 = 4