Solutions of chapter 16. Permutations | RD Sharma - Mathematics

1

Compute

(i)

(ii)

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

2

Prove that

3

Find x in each of the following :

3

Find x in each of the following :

3

Find x in each of the following :

4

Convert the following products into factorials :

5 ⋅ 6 ⋅ 7 ⋅ 8 ⋅ 9 ⋅ 10

4

Convert the following products into factorials :

3 ⋅ 6 ⋅ 9 ⋅ 12 ⋅ 15 ⋅ 18

4

Convert the following products into factorials :

(n + 1)(n + 2) (n + 3) …(2n)

4

Convert the following products into factorials :

1 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 9 …(2n – 1)

5

Which of the following are true :

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

5

Which of the following are true :

(2 × 3)! = 2! × 3!

6

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

7

If (n + 2)! = 60 [(n – 1)!], find n.

8

If (n + 1)! = 90 [(n – 1)!], find n.

9

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

10

If and are in the ratio 44 : 3, find n.

11

Prove that :

11

Prove that :

12

Prove that :

1

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?

2

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?

3

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?

4

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?

5

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

6

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

7

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

8

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?

9

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?

10

There are 5 books on Mathematics and 6 books on Physics in a book shop. In how many ways can a student buy:

(i) a Mathematics book and a Physics book

(ii) either a Mathematics book or a Physics book?

11

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

123

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?

13

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

14

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}?

15

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

16

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

17

How many three-digit numbers are there?

18

How many three-digit odd numbers are there?

19

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

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

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

20

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?

21

How many four-digit numbers can be formed with the digits 3, 5, 7, 8, 9 which are greater than 8000, if repetition of digits is not allowed?

22

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

23

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

24

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?

25

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

26

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?

27

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

28

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?

29

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?

30

A number lock on a suitcase has 3 wheels each labeled with ten digits 0 to 9. If the 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.

31

A customer forgets a four - digit 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 trails necessary to obtain the correct code.

32

In how many ways can get 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?

33

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

34

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?

35

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

36

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?

37

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

38

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

39

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?

40

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?

41

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

42

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

43

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

44

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

45

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

46

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

47

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 may gets all the prizes?

48

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.

1

Evaluate each of the following:

⁸P₃

1

Evaluate each of the following:

¹⁰P₄

1

Evaluate each of the following:

⁶P₆

1

Evaluate each of the following:

P(6, 4)

2

If P (5, r) = P (6, r – 1), find r.

3

If 5 P(4, n) = 6 P(5, n – 1), find n.

4

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

5

If ⁿP₄ = 360, find the value of n.

6

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

7

If P(11, r) = P (12, r – 1), find n.

8

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

9

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

10

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

11

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

12

Prove that:

1. P(1, 1) + 2. P(2, 2) + 3 . P(3, 3) + … + n . P(n, n) = P(n + 1, n + 1) – 1.

13

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

14

If find n.

15

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

16

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?

17

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?

18

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?

19

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

20

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

21

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

22

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

23

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?

24

. 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?

25

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

26

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

27

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?

28

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.

29

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

30

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

31

Find the numbers 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?

32

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.

1

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

2

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

the vowels come together?

2

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

the vowels never come together?

2

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

the vowels occupy only the odd places?

3

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

4

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

5

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?

6

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?

6

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

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

6

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

Are the vowels always together?

6

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

the vowels always occupy even places?

7

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

there is no restriction on letters?

7

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

each word begins with E?

7

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

ach word begins with O and ends with L?

7

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

all vowels come together?

7

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

all consonants come together?

8

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

9

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

10

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

11

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?

11

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

Are all letters used at a time?

11

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 a vowel?

12

How many three letter words can be made using the letters of the word ‘ORIENTAL.’

1

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

INDEPENDENCE

1

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

INTERMEDIATE

1

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

ARRANGE

1

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

INDIA

1

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

PAKISTAN

1

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

RUSSIA

1

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

SERIES

1

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

EXERCISES

1

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

CONSTANTINOPLE

2

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

3

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

4

Find the total number of arrangements of the letters in the expression a³ b² c⁴ when written at full length.

5

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

6

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

7

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?

8

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?

9

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

10

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

11

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

12

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

13

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

14

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

15

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

16

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?

17

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 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?

18

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

19

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

20

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

21

Find the total number of permutations of the letters of the word ‘INSTITUTE’.

22

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’.

23

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

24

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’.

25

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.

26

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.

27

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

i. the vowels always occupy even places?

ii. the relative order of vowels and consonants do not alter?

28

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’?

1

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

2

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

3

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

4

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.

5

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

6

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.

7

Write the number of words that can be formed out of the letters of the word COMMIT1EE’.

Note: The corrected question may be as follows:

Write the number of words that can be formed out of the letters of the word ‘COMMITTEE’.

8

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

9

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

10

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.

11

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

12

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

1