#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 7: Permutations and Combinations

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 7 Permutations and Combinations Solved Examples [Pages 115 - 121]

#### Short Answer

In a class, there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent the class for a function. In how many ways can the teacher make this selection?

How many numbers are there between 99 and 1000 having 7 in the units place?

How many numbers are there between 99 and 1000 having atleast one of their digits 7?

In how many ways can this diagram be coloured subject to the following two conditions?

(i) Each of the smaller triangle is to be painted with one of three colours: red, blue or green.

(ii) No two adjacent regions have the same colour.

In how many ways can 5 children be arranged in a line such that two particular children of them are always together

In how many ways can 5 children be arranged in a line such that two particular children of them are never together.

If all permutations of the letters of the word AGAIN are arranged in the order as in a dictionary. What is the 49th word?

In how many ways 3 mathematics books, 4 history books, 3 chemistry books and 2 biology books can be arranged on a shelf so that all books of the same subjects are together.

A student has to answer 10 questions, choosing atleast 4 from each of Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how many ways can the student choose 10 questions?

#### Long Answer

Suppose m men and n women are to be seated in a row so that no two women sit together. If m > n, show that the number of ways in which they can be seated is `(m!(m + 1)!)/((m - n + 1)1)`

Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are to be seated next to each other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together.

In a small village, there are 87 families, of which 52 families have atmost 2 children. In a rural development programme 20 families are to be chosen for assistance, of which atleast 18 families must have at most 2 children. In how many ways can the choice be made?

A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not want to borrow Mathematics Part II, unless Mathematics Part I is also borrowed. In how many ways can he choose the three books to be borrowed?

Find the number of permutations of n different things taken r at a time such that two specific things occur together.

#### Objective Type Questions

There are four bus routes between A and B; and three bus routes between B and C. A man can travel round-trip in number of ways by bus from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round trip?

72

144

14

19

In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7 men and 5 women?

45

350

4200

230

All the letters of the word ‘EAMCOT’ are arranged in different possible ways. The number of such arrangements in which no two vowels are adjacent to each other is ______.

360

144

72

54

Ten different letters of alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated is ______.

69760

30240

99748

99784

The number of signals that can be sent by 6 flags of different colours taking one or more at a time is ______.

63

1956

720

21

In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answer correct is ______.

11

12

27

63

The straight lines l_{1}, l_{2} and l_{3} are parallel and lie in the same plane. A total numbers of m points are taken on l1; n points on l_{2}, k points on l_{3}. The maximum number of triangles formed with vertices at these points are ______.

`""^((m + n + k))"C"_3`

`""^((m + n + k))"C"_3 - ""^n"C"_3 - ""^6"C"_3 - ""^k"C"_3`

^{m}C_{3}+^{n}C_{3}+^{k}C_{3}^{m}C_{3}×^{n}C_{3}×^{k}C_{3}

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 7 Permutations and Combinations Exercise [Pages 26 - 128]

#### Short Answer

Eight chairs are numbered 1 to 8. Two women and 3 men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4 and then men select from the remaining chairs. Find the total number of possible arrangements.

If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary. Then what is the rank of the word RACHIT?

A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. Find the number of different ways of doing question

Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the point

We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can selections be made?

How many committee of five persons with a chairperson can be selected from 12 persons.

How many automobile license plates can be made if each plate contains two different letters followed by three different digits?

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected from the lot.

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together

Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.

There are 10 persons named P_{1}, P_{2}, P_{3}, ... P_{10}. Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P1 must occur whereas P_{4} and P_{5} do not occur. Find the number of such possible arrangements.

There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.

A box contains two white, three black and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw

If ^{n}C_{r – 1} = 36, ^{n}C_{r} = 84 and ^{n}C_{r + 1} = 126, then find ^{r}C_{2}.

Find the number of integers greater than 7000 that can be formed with the digits 3, 5, 7, 8 and 9 where no digits are repeated.

If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, in how many points will they intersect each other?

In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64. How many telephone numbers have all six digits distinct?

In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.

A convex polygon has 44 diagonals. Find the number of its sides.

#### Long Answer

18 mice were placed in two experimental groups and one control group, with all groups equally large. In how many ways can the mice be placed into three groups?

A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if they can be of any colour

A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if two must be white and two red

A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if they must all be of the same colour.

In how many ways can a football team of 11 players be selected from 16 players? How many of them will include 2 particular players?

In how many ways can a football team of 11 players be selected from 16 players? How many of them will exclude 2 particular players?

A sports team of 11 students is to be constituted, choosing at least 5 from Class XI and atleast 5 from Class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has no girls

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has at least one boy and one girl

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has at least three girls.

#### Objective Type Questions from 26 to 40

If ^{n}C_{12} = ^{n}C_{8}, then n is equal to ______.

20

12

6

30

The number of possible outcomes when a coin is tossed 6 times is ______.

36

64

12

32

The number of different four-digit numbers that can be formed with the digits 2, 3, 4, 7 and using each digit only once is ______.

120

96

24

100

The sum of the digits in unit place of all the numbers formed with the help of 3, 4, 5 and 6 taken all at a time is ______.

432

108

36

18

Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to ______.

60

120

7200

720

A five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is ______.

216

600

240

3125

Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. The total number of persons in the room is ______.

11

12

13

14

The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is ______.

105

15

175

185

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is ______.

6

18

12

9

The number of ways in which a team of eleven players can be selected from 22 players always including 2 of them and excluding 4 of them is ______.

^{16}C_{11}^{16}C_{5}^{16}C_{9}^{20}C_{9}_{ }

The number of 5-digit telephone numbers having atleast one of their digits repeated is ______.

90,000

10,000

30,240

69,760

The number of ways in which we can choose a committee from four men and six women so that the committee includes at least two men and exactly twice as many women as men is ______.

94

126

128

None

The total number of 9 digit numbers which have all different digits is ______.

10!

9!

9 × 9!

10 × 10!

The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is ______.

1440

144

7!

^{4}C_{4}×^{3}C_{3}

Given 5 different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking at least one green and one blue dye is ______.

3600

3720

3800

3600

#### Fill in the Blanks 41 to 50

If ^{n}P_{r} = 840, ^{n}C_{r} = 35, then r = ______.

^{15}C_{8} + ^{15}C_{9} – ^{15}C_{6} – ^{15}C_{7} = ______.

The number of permutations of n different objects, taken r at a line, when repetitions are allowed, is ______.

The number of different words that can be formed from the letters of the word INTERMEDIATE such that two vowels never come together is ______.

Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls. The number of ways in which this can be done if at least 2 are red is ______.

The number of six-digit numbers, all digits of which are odd is ______.

In a football championship, 153 matches were played, Every two teams played one match with each other. The number of teams, participating in the championship is ______.

The total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two signs ‘–’ occur together is ______.

A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee ______.

A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ______.

#### State whether the following statement is True or False: 51 to 59

There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is ^{12}C_{2} – ^{5}C_{2}.

True

False

Three letters can be posted in five letterboxes in 35 ways.

True

False

In the permutations of n things, r taken together, the number of permutations in which m particular things occur together is `""^(n - m)"P"_(r - m) xx ""^r"P"_m`.

True

False

In a steamer there are stalls for 12 animals, and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in 3^{12} ways.

True

False

If some or all of n objects are taken at a time, the number of combinations is 2^{n} – 1.

True

False

There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls. It is being given that the balls of the same colour are identical.

True

False

Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on other side of the table. The number of ways in which the seating arrangements can be made is `(11!)/(5!6!) (9!)(9!)`.

True

False

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. He can choose the seven questions in 650 ways.

True

False

To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes. If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways in which the selection can be made is ^{5}C_{3} × ^{20}C_{9}.

True

False

#### Match the column C1 to C2 from 60 to 64:

There are 3 books on Mathematics, 4 on Physics and 5 on English. How many different collections can be made such that each collection consists of:

C_{1} |
C_{2} |

(a) One book of each subject; | (i) 3968 |

(b) At least one book of each subject: | (ii) 60 |

(c) At least one book of English: | (iii) 3255 |

Five boys and five girls form a line. Find the number of ways of making the seating arrangement under the following condition:

C_{1} |
C_{2} |

(a) Boys and girls alternate: | (i) 5! × 6! |

(b) No two girls sit together : | (ii) 10! – 5! 6! |

(c) All the girls sit together | (iii) (5!)^{2} + (5!)^{2} |

(d) All the girls are never together : | (iv) 2! 5! 5! |

There are 10 professors and 20 lecturers out of whom a committee of 2 professors and 3 lecturer is to be formed. Find:

C_{1} |
C_{2} |

(a) In how many ways committee: can be formed | (i) ^{10}C_{2} × ^{19}C_{3} |

(b) In how many ways a particular: professor is included | (ii) ^{10}C_{2} × ^{19}C2 |

(c) In how many ways a particular: lecturer is included | (iii) ^{9}C_{1} × ^{20}C_{3} |

(d) In how many ways a particular: lecturer is excluded | (iv) ^{10}C_{2} × ^{20}C_{3} |

Using the digits 1, 2, 3, 4, 5, 6, 7, a number of 4 different digits is formed. Find

C_{1} |
C_{2} |

(a) How many numbers are formed? | (i) 840 |

(b) How many number are exactly divisible by 2? | (i) 200 |

(c) How many numbers are exactly divisible by 25? | (iii) 360 |

(d) How many of these are exactly divisible by 4? | (iv) 40 |

How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if

C_{1} |
C_{2} |

(a) 4 letters are used at a time | (i) 720 |

(b) All letters are used at a time | (ii) 240 |

(c) All letters are used but the first is a vowel | (iii) 360 |

## Chapter 7: Permutations and Combinations

## NCERT solutions for Mathematics Exemplar Class 11 chapter 7 - Permutations and Combinations

NCERT solutions for Mathematics Exemplar Class 11 chapter 7 (Permutations and Combinations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 7 Permutations and Combinations are Combination, Permutations, Fundamental Principles of Counting, Introduction of Permutations and Combinations, Permutation Formula to Rescue and Type of Permutation, Smaller Set from Bigger Set, Derivation of Formulae and Their Connections, Simple Applications of Permutations and Combinations, Factorial N (N!) Permutations and Combinations.

Using NCERT Class 11 solutions Permutations and Combinations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

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