#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 16 : Permutations

#### Pages 4 - 5

Compute:

(i)\[\frac{30!}{28!}\]

Compute:

Compute:

L.C.M. (6!, 7!, 8!)

Prove that

Find *x* in each of the following:

Find *x* in each of the following:

Find *x* in each of the following:

Convert the following products into factorials:

5 · 6 · 7 · 8 · 9 · 10

Convert the following products into factorials:

3 · 6 · 9 · 12 · 15 · 18

Convert the following products into factorials:

(*n* + 1) (*n* + 2) (*n* + 3) ... (2*n*)

Convert the following products into factorials:

1 · 3 · 5 · 7 · 9 ... (2*n* − 1)

Which of the following are true:

(2 +3)! = 2! + 3!

Which of the following are true:

(2 × 3)! = 2! × 3!

Prove that: *n*! (*n* + 2) = *n*! + (*n* + 1)!

If (*n* + 2)! = 60 [(*n* − 1)!], find *n. *

If (n + 1)! = 90 [(n − 1)!], find *n*.

If (*n* + 3)! = 56 [(*n* + 1)!], find *n*.

If \[\frac{(2n)!}{3! (2n - 3)!}\] and \[\frac{n!}{2! (n - 2)!}\] are in the ratio 44 : 3, find *n*.

Prove that:

*n*(

*n*− 1) (

*n*− 2) ... (

*n*− (

*r*− 1))

Prove that:

\[\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}\]

Prove that:

^{n}[1 · 3 · 5 ... (2

*n*− 1) (2

*n*+ 1)]

#### Pages 14 - 16

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

A person wants to buy one fountain pen, one ball pen and one pencil from a stationery shop. If there are 10 fountain pen varieties, 12 ball pen varieties and 5 pencil varieties, in how many ways can he select these articles?

From Goa to Bombay there are two roots; air, and sea. From Bombay to Delhi there are three routes; air, rail and road. From Goa to Delhi via Bombay, how many kinds of routes are there?

A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of calendars should it prepare to serve for all the possibilities in future years?

There are four parcels and five post-offices. In how many different ways can the parcels be sent by registered post?

A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?

In how many ways can an examinee answer a set of ten true/false type questions?

A letter lock consists of three rings each marked with 10 different letters. In how many ways it is possible to make an unsuccessful attempt to open the lock?

There are 6 multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three have 2 each?

There are 5 books on Mathematics and 6 books on Physics in a book shop. In how many ways can a students buy : (i) a Mathematics book and a Physics book (ii) either a Mathematics book or a Physics book?

Given 7 flags of different colours, how many different signals can be generated if a signal requires the use of two flags, one below the other?

A team consists of 6 boys and 4 girls and other has 5 boys and 3 girls. How many single matches can be arranged between the two teams when a boy plays against a boy and a girl plays against a girl?

Twelve students complete in a race. In how many ways first three prizes be given?

How many A.P.'s with 10 terms are there whose first term is in the set {1, 2, 3} and whose common difference is in the set {1, 2, 3, 4, 5}?

From among the 36 teachers in a college, one principal, one vice-principal and the teacher-incharge are to be appointed. In how many ways can this be done?

How many three-digit numbers are there with no digit repeated?

How many three-digit numbers are there?

How many three-digit odd numbers are there?

How many different five-digit number licence plates can be made if

first digit cannot be zero and the repetition of digits is not allowed,

How many different five-digit number licence plates can be made if

the first-digit cannot be zero, but the repetition of digits is allowed?

How many four-digit numbers can be formed with the digits 3, 5, 7, 8, 9 which are greater than 7000, if repetition of digits is not allowed?

Since the number has to be greater than 8000, the thousand's place can be filled by only two digits, i.e. 8 and 9.

Now, the hundred's place can be filled with the remaining 4 digits as the repetition of the digits is not allowed.

The ten's place can be filled with the remaining 3 digits.

The unit's place can be filled with the remaining 2 digits.

Total numbers that can be formed = `2xx4xx3xx2=48`

In how many ways can six persons be seated in a row?

How many 9-digit numbers of different digits can be formed?

How many odd numbers less than 1000 can be formed by using the digits 0, 3, 5, 7 when repetition of digits is not allowed?

How many 3-digit numbers are there, with distinct digits, with each digit odd?

How many different numbers of six digits each can be formed from the digits 4, 5, 6, 7, 8, 9 when repetition of digits is not allowed?

How many different numbers of six digits can be formed from the digits 3, 1, 7, 0, 9, 5 when repetition of digits is not allowed?

How many four digit different numbers, greater than 5000 can be formed with the digits 1, 2, 5, 9, 0 when repetition of digits is not allowed?

Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?

A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.

A customer forgets a four-digits code for an Automatic Teller Machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6 and 9. Find the largest possible number of trials necessary to obtain the correct code.

In how many ways can three jobs I, II and III be assigned to three persons *A*, *B* and *C* if one person is assigned only one job and all are capable of doing each job?

How many natural numbers not exceeding 4321 can be formed with the digits 1, 2, 3 and 4, if the digits can repeat?

How many numbers of six digits can be formed from the digits 0, 1, 3, 5, 7 and 9 when no digit is repeated? How many of them are divisible by 10 ?

If three six faced die each marked with numbers 1 to 6 on six faces, are thrown find the total number of possible outcomes ?

A coin is tossed three times and the outcomes are recorded. How many possible outcomes are there? How many possible outcomes if the coin is tossed four times? Five times? *n* times?

How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5 if the digits can be repeated in the same number?

How many three digit numbers can be formed by using the digits 0, 1, 3, 5, 7 while each digit may be repeated any number of times?

How many natural numbers less than 1000 can be formed from the digits 0, 1, 2, 3, 4, 5 when a digit may be repeated any number of times?

How many five digit telephone numbers can be constructed using the digits 0 to 9. If each number starts with 67 and no digit appears more than once?

Find the number of ways in which 8 distinct toys can be distributed among 5 childrens.

Find the number of ways in which one can post 5 letters in 7 letter boxes ?

Three dice are rolled. Find the number of possible outcomes in which at least one die shows 5 ?

Find the total number of ways in which 20 balls can be put into 5 boxes so that first box contains just one ball ?

In how many ways can 5 different balls be distributed among three boxes?

In how many ways can 7 letters be posted in 4 letter boxes?

In how many ways can 4 prizes be distributed among 5 students, when

(i) no student gets more than one prize?

(ii) a student may get any number of prizes?

(iii) no student gets all the prizes?

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 ?

#### Pages 28 - 29

Evaluate each of the following:

^{8}P_{3}

Evaluate each of the following:

^{10}P

_{4 }

Evaluate each of the following:

^{6}P_{6 }

Evaluate each of the following:

P(6, 4)

If P (5, r) = P (6, r − 1), find r ?

If 5 P(4, n) = 6. P (5, n − 1), find n ?

If P (n, 5) = 20. P(n, 3), find n ?

If ^{n}P_{4} = 360, find the value of n.

If *P* (9, *r*) = 3024, find *r*.

If P(11, r) = P (12, r − 1) find r.

If P (n, 4) = 12 . P (n, 2), find n.

If P (n − 1, 3) : P (n, 4) = 1 : 9, find n.

If P (2n − 1, n) : P (2n + 1, n − 1) = 22 : 7 find n.

If P (n, 5) : P (n, 3) = 2 : 1, find n.

Prove that:1 . *P* (1, 1) + 2 . *P* (2, 2) + 3 . *P* (3, 3) + ... + *n* . *P* (*n*, *n*) = *P* (*n* + 1, *n* + 1) − 1.

If P (15, r − 1) : P (16, r − 2) = 3 : 4, find r.

If ^{n +}^{5}P_{n}_{ +1 }=\[\frac{11 (n - 1)}{2}\]^{n +}^{3}P* _{n}*, find

*n*.

In how many ways can five children stand in a queue?

From among the 36 teachers in a school, one principal and one vice-principal are to be appointed. In how many ways can this be done?

Four letters E, K, S and V, one in each, were purchased from a plastic warehouse. How many ordered pairs of letters, to be used as initials, can be formed from them?

Four books, one each in Chemistry, Physics, Biology and Mathematics, are to be arranged in a shelf. In how many ways can this be done?

Find the number of different 4-letter words, with or without meanings, that can be formed from the letters of the word 'NUMBER'.

How many three-digit numbers are there, with distinct digits, with each digit odd?

How many words, with or without meaning, can be formed by using all the letters of the word 'DELHI', using each letter exactly once?

How many words, with or without meaning, can be formed by using the letters of the word 'TRIANGLE'?

There are two works each of 3 volumes and two works each of 2 volumes; In how many ways can the 10 books be placed on a shelf so that the volumes of the same work are not separated?

There are 6 items in column A and 6 items in column B. A student is asked to match each item in column A with an item in column B. How many possible, correct or incorrect, answers are there to this question?

How many three-digit numbers are there, with no digit repeated?

How many 6-digit telephone numbers can be constructed with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each number starts with 35 and no digit appears more than once?

In how many ways can 6 boys and 5 girls be arranged for a group photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?

If *a* denotes the number of permutations of (*x* + 2) things taken all at a time, b the number of permutations of *x* things taken 11 at a time and *c* the number of permutations of *x* − 11 things taken all at a time such that *a* = 182 *bc*, find the value of *x*.

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

How many 3-digit even number can be made using the digits 1, 2, 3, 4, 5, 6, 7, if no digits is repeated?

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repeated? How many of these will be even?

All the letters of the word 'EAMCOT' are arranged in different possible ways. Find the number of arrangements in which no two vowels are adjacent to each other.

#### Pages 36 - 37

In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?

In how many ways can the letters of the word 'STRANGE' be arranged so that

the vowels come together?

In how many ways can the letters of the word 'STRANGE' be arranged so that

the vowels never come together?

In how many ways can the letters of the word 'STRANGE' be arranged so that

the vowels occupy only the odd places?

How many words can be formed from the letters of the word 'SUNDAY'? How many of these begin with D?

How many words can be formed out of the letters of the word, 'ORIENTAL', so that the vowels always occupy the odd places?

How many different words can be formed with the letters of word 'SUNDAY'? How many of the words begin with N? How many begin with N and end in Y?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the vowels are always together?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the letter G always occupies the first place?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

the letters P and I respectively occupy first and last place?

the vowels always occupy even places?

How many permutations can be formed by the letters of the word, 'VOWELS', when

there is no restriction on letters?

How many permutations can be formed by the letters of the word, 'VOWELS', when

each word begins with E?

How many permutations can be formed by the letters of the word, 'VOWELS', when

each word begins with O and ends with L?

How many permutations can be formed by the letters of the word, 'VOWELS', when

all vowels come together?

How many permutations can be formed by the letters of the word, 'VOWELS', when

all consonants come together?

How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?

In how many ways can a lawn tennis mixed double be made up from seven married couples if no husband and wife play in the same set?

*m* men and *n* women are to be seated in a row so that no two women sit together. if *m* > *n* then show that the number of ways in which they can be seated as\[\frac{m! (m + 1)!}{(m - n + 1) !}\]

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

4 letters are used at a time?

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

all letters are used at a time.

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

all letters are used but first is vowel.

How many three letter words can be made using the letters of the word 'ORIENTAL'?

#### Pages 42 - 44

Find the number of words formed by permuting all the letters of the following words:

INDEPENDENCE

Find the number of words formed by permuting all the letters of the following words:

INTERMEDIATE

Find the number of words formed by permuting all the letters of the following words:

ARRANGE

Find the number of words formed by permuting all the letters of the following words:

INDIA

Find the number of words formed by permuting all the letters of the following words:

PAKISTAN

Find the number of words formed by permuting all the letters of the following words:

RUSSIA

Find the number of words formed by permuting all the letters of the following words:

SERIES

Find the number of words formed by permuting all the letters of the following words:

EXERCISES

Find the number of words formed by permuting all the letters of the following words:

CONSTANTINOPLE

In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?

How many words can be formed with the letters of the word 'UNIVERSITY', the vowels remaining together?

Find the total number of arrangements of the letters in the expression *a*^{3} *b*^{2} *c*^{4} when written at full length.

How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?

How many words can be formed by arranging the letters of the word 'MUMBAI' so that all M's come together?

How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?

How many different signals can be made from 4 red, 2 white and 3 green flags by arranging all of them vertically on a flagstaff?

How many number of four digits can be formed with the digits 1, 3, 3, 0?

In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?

How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.

How many words can be formed from the letters of the word 'SERIES' which start with S and end with S?

How many permutations of the letters of the word 'MADHUBANI' do not begin with M but end with I?

Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.

There are three copies each of 4 different books. In how many ways can they be arranged in a shelf?

How many different arrangements can be made by using all the letters in the word 'MATHEMATICS'. How many of them begin with C? How many of them begin with T?

A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initials *A* (for Adenine), *C* (for Cytosine), *G* (for Guanine) and *T* (for Thymine) and 3 molecules of each kind. How many different such arrangements are possible?

In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?

How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?

Find the total number of permutations of the letters of the word 'INSTITUTE'.

The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word 'SURITI'.

If the letters of the word 'LATE' be permuted and the words so formed be arranged as in a dictionary, find the rank of the word LATE.

If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'.

If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation debac ?

Find the total number of ways in which six '+' and four '−' signs can be arranged in a line such that no two '−' signs occur together.

In how many ways can the letters of the word

"INTERMEDIATE" be arranged so that:the vowels always occupy even places?

In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:

the relative order of vowels and consonants do not alter?

The letters of the word 'ZENITH' are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word 'ZENITH'?

#### Page 45

In how many ways can 4 letters be posted in 5 letter boxes?

Write the number of 5 digit numbers that can be formed using digits 0, 1 and 2 ?

In how many ways 4 women draw water from 4 taps, if no tap remains unused?

Write the total number of possible outcomes in a throw of 3 dice in which at least one of the dice shows an even number.

Write the number of arrangements of the letters of the word BANANA in which two N's come together.

Write the number of ways in which 7 men and 7 women can sit on a round table such that no two women sit together ?

Write the number of words that can be formed out of the letters of the word 'COMMITTEE' ?

Write the number of all possible words that can be formed using the letters of the word 'MATHEMATICS'.

Write the number of ways in which 6 men and 5 women can dine at a round table if no two women sit together ?

Write the number of ways in which 5 boys and 3 girls can be seated in a row so that each girl is between 2 boys ?

Write the remainder obtained when 1! + 2! + 3! + ... + 200! is divided by 14 ?

Write the number of numbers that can be formed using all for digits 1, 2, 3, 4 ?

#### Pages 46 - 47

The number of permutations of *n* different things taking *r* at a time when 3 particular things are to be included is

^{n − }^{3}*P _{r}*

_{− 3}

^{n − }^{3}*P _{r}*

^{n}P_{r}_{ − 3}

*r* ! ^{n − }^{3}*C*_{r}_{− 3}

The number of five-digit telephone numbers having at least one of their digits repeated is

90000

100000

30240

69760

The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is

574

36

754

144

The number of different signals which can be given from 6 flags of different colours taking one or more at a time, is

1958

1956

16

64

The number of words from the letters of the word 'BHARAT' in which B and H will never come together, is

360

240

120

none of these.

The number of six letter words that can be formed using the letters of the word "ASSIST" in which S's alternate with other letters is

12

24

18

none of these.

The number of ways in which the letters of the word 'CONSTANT' can be arranged without changing the relative positions of the vowels and consonants is

360

256

444

none of these.

Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and divisible by 4 is

24

30

125

100

If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is

324

341

359

none of these

If in a group of *n* distinct objects, the number of arrangements of 4 objects is 12 times the number of arrangements of 2 objects, then the number of objects is

10

8

6

none of these.

The number of ways in which 6 men can be arranged in a row so that three particular men are consecutive, is

4! × 3!

4!

3! × 3!

none of these.

A 5-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is

216

600

240

3125

The product of *r* consecutive positive integers is divisible by

*r* !

(*r* − 1) !

(*r* + 1) !

none of these.

If ^{k + }^{5}*P _{k}*

_{ + 1}=\[\frac{11 (k - 1)}{2}\].

^{k + }

^{3}

*P*

_{k}, then the values of

*k*are

7 and 11

6 and 7

2 and 11

2 and 6

The number of arrangements of the letters of the word BHARAT taking 3 at a time is

72

120

14

none of these.

The number of words that can be made by re-arranging the letters of the word APURBA so that vowels and consonants are alternate is

18

35

36

none of these

The number of different ways in which 8 persons can stand in a row so that between two particular persons *A* and *B* there are always two persons, is

60 × 5!

15 × 4! × 5!

4! × 5!

none of these.

The number of ways in which the letters of the word ARTICLE can be arranged so that even places are always occupied by consonants is

576

^{4}*C*_{3} × 4!

2 × 4!

none of these.

In a room there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amounts of illumination is

12^{2} − 1

2^{12}

2^{12} − 1

none of these

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 16 - Permutations

RD Sharma solutions for Class 11 Maths chapter 16 (Permutations) 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 Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 16 Permutations are Factorial N (N!) Permutations and Combinations, Simple Applications of Permutations and Combinations, Derivation of Formulae and Their Connections, Smaller Set from Bigger Set, Permutation Formula to Rescue and Type of Permutation, Introduction of Permutations and Combinations, Concept of Combinations, Concept of Permutations, Fundamental Principle of Counting.

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