#### Chapters

## Chapter 3: Permutations and Combination

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Exercise 3.1 [Pages 47 - 48]

A teacher wants to select the class monitor in a class of 30 boys and 20 girls. In how many ways can the monitor be selected if the monitor must be a girl or a boy?

A Signal is generated from 2 flags by putting one flag above the other. If 4 flags of different colours are available, how many different signals can be generated?

How many two letter words can be formed using letters from the word SPACE, when repetition of letters is allowed?

How many two letter words can be formed using letters from the word SPACE, when repetition of letters is not allowed?

How many three-digit numbers can be formed from the digits 0, 1, 3, 5, 6 if repetitions of digits are allowed?

How many three-digit numbers can be formed from the digits 0, 1, 3, 5, 6 if repetitions of digits are not allowed?

How many three-digit numbers can be formed using the digits 2, 3, 4, 5, 6 if digits can be repeated?

A letter lock contains 3 rings, each ring containing 5 different letters. Determine the maximum number of false trials that can be made before the lock is opened?

In a test, 5 questions are of the form 'state, true or false'. No student has got all answers correct. Also, the answer of every student is different. Find the number of students appeared for the test.

How many numbers between 100 and 1000 have 4 in the units place?

How many numbers between 100 and 1000 have the digit 7 exactly once?

How many four digit numbers will not exceed 7432 if they are formed using the digits 2, 3, 4, 7 without repetition?

If numbers are formed using digits 2, 3, 4, 5, 6 without repetition, how many of them will exceed 400?

How many numbers formed with the digits 0, 1, 2, 5, 7, 8 will fall between 13 and 1000 if digits can be repeated?

A school has three gates and four staircases from the first floor to the second floor. How many ways does a student have to go from outside the school to his classroom on the second floor?

How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 are divisible by 5 if digits are not repeated?

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Exercise 3.2 [Pages 49 - 50]

Evaluate: 8!

Evaluate: 10!

Evaluate: 10! – 6!

Evaluate: (10 – 6)!

Compute: `(12!)/(6!)`

Compute: `(12/6)!`

Compute: (3 × 2)!

Compute: 3! × 2!

Compute: `(9!)/(3! 6!)`

Compute: `(6! - 4!)/(4!)`

Compute: `(8!)/(6! - 4!)`

Compute: `(8!)/((6 - 4)!)`

Write in terms of factorial.

5 × 6 × 7 × 8 × 9 × 10

Write in terms of factorial.

3 × 6 × 9 × 12 × 15

Write in terms of factorial.

6 × 7 × 8 × 9

Write in terms of factorial.

5 × 10 × 15 × 20

Evaluate : `("n"!)/("r"!("n" - "r")!)` for n = 8, r = 6

Evaluate : `("n"!)/("r"!("n" - "r")!)` for n = 12, r = 12

Evaluate : `("n"!)/("r"!("n" - "r")!)` for n = 15, r = 10

Evaluate : `("n"!)/("r"!("n" - "r")!)` for n = 15, r = 8

Find n, if `"n"/(8!) = 3/(6!) + (1!)/(4!)`

Find n, if `"n"/(6!) = 4/(8!) + 3/(6!)`

Find n, if `(1!)/("n"!) = (1!)/(4!) - 4/(5!)`

Find n, if (n + 1)! = 42 × (n – 1)!

Find n, if (n + 3)! = 110 × (n + 1)!

Find n, if: `((17 - "n")!)/((14 - "n")!)` = 5!

Find n, if: `((15 - "n")!)/((13 - "n")!)` = 12

Find n, if: `("n"!)/(3!("n" - 3)!) : ("n"!)/(5!("n" - 5)!)` = 5 : 3

Find n, if: `("n"!)/(3!("n" - 3)!) : ("n"!)/(5!("n" - 7)!)` = 1 : 6

Find n, if: `((2"n")!)/(7!(2"n" - 7)!) : ("n"!)/(4!("n" - 4)!)` = 24 : 1

Show that `("n"!)/("r"!("n" - "r")!) + ("n"!)/(("r" - 1)!("n" - "r" + 1)!) = (("n" + 1)!)/("r"!("n" - "r" + 1)!)`

Show that `(9!)/(3!6!) + (9!)/(4!5!) = (10!)/(4!6!)`

Show that `((2"n")!)/("n"!)` = 2^{n} (2n – 1)(2n – 3) ... 5.3.1

Simplify `((2"n" + 2)!)/((2"n")!)`

Simplify `(("n" + 3)!)/(("n"^2 - 4)("n" + 1)!)`

Simplify `1/("n"!) - 1/(("n" - 1)!) - 1/(("n" - 2)!)`

Simplify n[n! + (n – 1)!] + n^{2}(n – 1)! + (n + 1)!

Simplify `("n" + 2)/("n"!) - (3"n" + 1)/(("n" + 1)!)`

Simplify `1/(("n" - 1)!) + (1 - "n")/(("n" + 1)!)`

Simplify `1/("n"!) - 3/(("n" + 1)!) - ("n"^2 - 4)/(("n" + 2)!)`

Simplify `("n"^2 - 9)/(("n" + 3)!) + 6/(("n" + 2)!) - 1/(("n" + 1)!)`

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Exercise 3.3 [Pages 54 - 55]

Find n, if ^{n}P_{6} : ^{n}P_{3} = 120 : 1

Find m and n, if ^{(m+n)}P_{2} = 56 and ^{(m-n)}P_{2} = 12

Find r, if ^{12}P_{r–2} : ^{11}P_{r–1} = 3 : 14

Show that (n + 1) (^{n}P_{r}) = (n – r + 1) [^{(n+1)}P_{r}]

How many 4 letter words can be formed using letters in the word MADHURI if letters can be repeated

How many 4 letter words can be formed using letters in the word MADHURI if letters cannot be repeated

Determine the number of arrangements of letters of the word ALGORITHM if vowels are always together

Determine the number of arrangements of letters of the word ALGORITHM if no two vowels are together

Determine the number of arrangements of letters of the word ALGORITHM if consonants are at even positions

Determine the number of arrangements of letters of the word ALGORITHM if O is the first and T is the last letter

In a group photograph, 6 teachers are in the first row and 18 students are in the second row. There are 12 boys and 6 girls among the students. If the middle position is reserved for the principal and if no two girls are together, find the number of arrangements.

Find the number of ways so that letters of the word HISTORY can be arranged as Y and T are together

Find the number of ways so that letters of the word HISTORY can be arranged as Y is next to T

Find the number of ways so that letters of the word HISTORY can be arranged as there is no restriction

Find the number of ways so that letters of the word HISTORY can be arranged as begin and end with vowel

Find the number of ways so that letters of the word HISTORY can be arranged as end in ST

Find the number of ways so that letters of the word HISTORY can be arranged as begin with S and end with T

Find the number of arrangements of the letters in the word SOLAPUR so that consonants and vowels are placed alternately

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 4, 5, 6, 8 if digits can be repeated

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 4, 5, 6, 8 if digits cannot be repeated

How many numbers can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition so that resulting numbers are between 100 and 1000?

Find the number of 6-digit numbers using the digits 3, 4, 5, 6, 7, 8 without repetition. How many of these numbers are divisible by 5

Find the number of 6-digit numbers using the digits 3, 4, 5, 6, 7, 8 without repetition. How many of these numbers are not divisible by 5

A code word is formed by two different English letters followed by two non-zero distinct digits. Find the number of such code words. Also, find the number of such code words that end with an even digit.

Find the number of ways in which 5 letters can be posted in 3 post boxes if any number of letters can be posted in a post box.

Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object always occurs

Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object never occurs

In how many ways can 5 different books be arranged on a shelf if there are no restrictions

In how many ways can 5 different books be arranged on a shelf if 2 books are always together

In how many ways can 5 different books be arranged on a shelf if 2 books are never together

3 boys and 3 girls are to sit in a row. How many ways can this be done if there are no restrictions

3 boys and 3 girls are to sit in a row. How many ways can this be done if there is a girl at each end

3 boys and 3 girls are to sit in a row. How many ways can this be done if boys and girls are at alternate places

3 boys and 3 girls are to sit in a row. How many ways can this be done if all boys sit together

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Exercise 3.4 [Page 57]

Find the number of permutations of letters in the following word:

DIVYA

Find the number of permutations of letters in the following word:

SHANTARAM

Find the number of permutations of letters in the following word:

REPRESENT

Find the number of permutations of letters in the following word:

COMBINE

Find the number of permutations of letters in the following word:

BALBHARATI

You have 2 identical books on English, 3 identical books on Hindi, and 4 identical books on Mathematics. Find the number of distinct ways of arranging them on a shelf

A coin is tossed 8 times. In how many ways can we obtain 4 heads and 4 tails?

A coin is tossed 8 times. In how many ways can we obtain at least 6 heads?

A bag has 5 red, 4 blue, and 4 green marbles. If all are drawn one by one and their colours are recorded, how many different arrangements can be found?

Find the number of ways of arranging letters of the word MATHEMATICAL How many of these arrangements have all vowels together?

Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have letters R and H never together?

Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have all vowels together?

How many different words are formed if the letters R is used thrice and letters S and T are used twice each?

Find the number of arrangements of letters in the word MUMBAI so that the letter B is always next to A

Find the number of arrangements of letters in the word CONSTITUTION that begin and end with N

Find the number of different ways of arranging letters in the word ARRANGE. How many of these arrangements do not have the two R’s and two A’s together?

How many distinct 5 digit numbers can be formed using the digits 3, 2, 3, 2, 4, 5

Find the number of distinct numbers formed using the digits 3, 4, 5, 6, 7, 8, 9, so that odd positions are occupied by odd digits

How many different 6-digit numbers can be formed using digits in the number 659942? How many of them are divisible by 4?

Find the number of distinct words formed from letters in the word INDIAN. How many of them have the two N’s together?

Find the number of different ways of arranging letters in the word PLATOON if the two O’s are never together

Find the number of different ways of arranging letters in the word PLATOON if consonants and vowels occupy alternate positions

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Exercise 3.5 [Page 61]

In how many different ways can 8 friends sit around a table?

A party has 20 participants. Find the number of distinct ways for the host to sit with them around a circular table. How many of these ways have two specified persons on either side of the host?

Delegates from 24 countries participate in a round table discussion. Find the number of seating arrangments where two specified delegates are always together

Delegates from 24 countries participate in a round table discussion. Find the number of seating arrangments where two specified delegates are never together

Find the number of ways for 15 people to sit around the table so that no two arrangements have the same neighbours

A committee of 10 members sits around a table. Find the number of arrangements that have the president and the vice-president together.

Five men, two women, and a child sit around a table. Find the number of arrangements where the child is seated between the two women

Five men, two women, and a child sit around a table. Find the number of arrangements where the child is seated between two men

Eight men and six women sit around a table. How many of sitting arrangements will have no two women together?

Find the number of seating arrangements for 3 men and 3 women to sit around a table so that exactly two women are together

Four objects in a set of ten objects are alike. Find the number of ways of arranging them in a circular order

Fifteen persons sit around a table. Find the number of arrangements that have two specified persons not sitting side by side.

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Exercise 3.6 [Pages 64 - 66]

Find the value of ^{15}C_{4}

Find the value of ^{80}C_{2}

Find the value of ^{15}C_{4} + ^{15}C_{5}

Find the value of ^{20}C_{16} – ^{19}C_{16}

Find n if ^{6}P_{2} = n ^{6}C_{2}

Find n if ^{2n}C_{3} : ^{n}C_{2} = 52 : 3

Find n if ^{n}C_{n–3} = 84

Find r if ^{14}C_{2r} : ^{10}C_{2r–4} = 143 : 10

Find n and r if ^{n}P_{r} = 720 and ^{n}C_{n–r} = 120

Find n and r if ^{n}C_{r–1} : ^{n}C_{r} : ^{n}C_{r+1} = 20 : 35 : 42

If ^{n}P_{r} = 1814400 and ^{n}C_{r} = 45, find ^{n+4}C_{r+3}

If ^{n}C_{r–1} = 6435, ^{n}C_{r} = 5005, ^{n}C_{r+1} = 3003, find ^{r}C_{5}

Find the number of ways of drawing 9 balls from a bag that has 6 red balls, 8 green balls, and 7 blue balls so that 3 balls of every colour are drawn

Find the number of ways of selecting a team of 3 boys and 2 girls from 6 boys and 4 girls

After a meeting, every participant shakes hands with every other participants. If the number of handshakes is 66, find the number of participants in the meeting.

If 20 points are marked on a circle, how many chords can be drawn?

Find the number of diagonals of an n-sided polygon. In particular, find the number of diagonals when n = 10

Find the number of diagonals of an n-sided polygon. In particular, find the number of diagonals when n = 15

Find the number of diagonals of an n-sided polygon. In particular, find the number of diagonals when n = 12

Find the number of diagonals of an n-sided polygon. In particular, find the number of diagonals when n = 8

There are 20 straight lines in a plane so that no two lines are parallel and no three lines are concurrent. Determine the number of points of intersection

Ten points are plotted on a plane. Find the number of straight lines obtained by joining these points if no three points are collinear

Ten points are plotted on a plane. Find the number of straight lines obtained by joining these points if four points are collinear

Find the number of triangles formed by joining 12 points if no three points are collinear

Find the number of triangles formed by joining 12 points if four points are collinear

A word has 8 consonants and 3 vowels. How many distinct words can be formed if 4 consonants and 2 vowels are chosen?

Find n if ^{n}C_{8} = ^{n}C_{12}

Find n if ^{23}C_{3n} = ^{23}C_{2n+3}

Find n if ^{21}C_{6n} = `""^21"C"_(("n"^2 + 5))`

Find n if ^{2n}C_{r–1} = ^{2n}C_{r+1}

Find n if ^{n}C_{n–2} = 15

Find x if ^{n}P_{r} = x ^{n}C_{r }

Find r if ^{11}C_{4} + ^{11}C_{5} + ^{12}C_{6} + ^{13}C_{7} = ^{14}C_{r}

Find the value of `sum_("r" = 1)^4 ""^((21 - "r"))"C"_4`

Find the differences between the greatest values in the following:

^{14}C_{r} and ^{12}C_{r}

Find the differences between the greatest values in the following:

^{13}C_{r} and ^{8}C_{r}

Find the differences between the greatest values in the following:

^{15}C_{r} and ^{11}C_{r}

In how many ways can a boy invite his 5 friends to a party so that at least three join the party?

A group consists of 9 men and 6 women. A team of 6 is to be selected. How many of possible selections will have at least 3 women?

A committee of 10 persons is to be formed from a group of 10 women and 8 men. How many possible committees will have at least 5 women? How many possible committees will have men in majority?

A question paper has two sections. section I has 5 questions and section II has 6 questions. A student must answer at least two question from each section among 6 questions he answers. How many different choices does the student have in choosing questions?

There are 3 wicketkeepers and 5 bowlers among 22 cricket players. A team of 11 players is to be selected so that there is exactly one wicketkeeper and at least 4 bowlers in the team. How many different teams can be formed?

Five students are selected from 11. How many ways can these students be selected if two specified students are selected?

Five students are selected from 11. How many ways can these students be selected if two specified students are not selected?

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Miscellaneous Exercise 3 [Page 67]

Select the correct answer from the given alternatives.

A college offers 5 courses in the morning and 3 in the evening. The number of ways a student can select exactly one course, either in the morning or in the evening

5

3

8

15

Select the correct answer from the given alternatives.

A college has 7 courses in the morning and 3 in the evening. The possible number of choices with the student if he wants to study one course in the morning and one in the evening is -

21

4

42

10

Select the correct answer from the given alternatives.

In how many ways can 8 Indians and, 4 American and 4 Englishmen can be seated in a row so that all person of the same nationality sit together?

3! 8!

3! 4! 8! 4!

4! 4!

8! 4! 4!

Select the correct answer from the given alternatives.

In how many ways can 10 examination papers be arranged so that the best and the worst papers never come together?

9 × 8!

8 × 8!

9 × 9!

8 × 9!

Select the correct answer from the given alternatives.

In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate

12

288

144

256

Select the correct answer from the given alternatives.

Find the number of triangles which can be formed by joining the angular points of a polygon of 8 sides as vertices.

16

56

24

8

Select the correct answer from the given alternatives.

A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 8 from part A and 5 from part B, In how many ways can he choose the questions?

320

750

40

11340

Select the correct answer from the given alternative

There are 10 persons among whom two are brothers. The total number of ways in which these persons can be seated around a round table so that exactly one person sits between the brothers is equal to:

2! × 7!

2! × 8!

3! × 7!

3! × 8!

Select the correct answer from the given alternatives.

The number of arrangements of the letters of the word BANANA in which two N's do not appear adjacently

80

60

40

100

Select the correct answer from the given alternatives.

The number of ways in which 5 male and 2 female members of a committee can be seated around a round table so that the two females are not seated together is

840

600

720

480

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Permutations and Combination Miscellaneous Exercise 3 [Page 68]

Answer the following:

Find the value of r if ^{56}P_{r+6} : ^{54}P_{r+3} = 30800 : 1

Answer the following:

How many words can be formed by writing letters in the word CROWN in different order?

Answer the following:

Find the number of words that can be formed by using all the letters in the word REMAIN If these words are written in dictionary order, what will be the 40^{th} word?

Answer the following:

Capital English alphabet has 11 symmetric letters that appear same when looked at in a mirror. These letters are A, H, I, M, O, T, U, V, W, X, and Y. How many symmetric three letter passwords can be formed using these letters?

Answer the following:

How many numbers formed using the digits 3, 2, 0, 4, 3, 2, 3 exceed one million?

Answer the following:

Ten students are to be selected for a project from a class of 30 students. There are 4 students who want to be together either in the project or not in the project. Find the number of possible selections

Answer the following:

A student finds 7 books of his interest but can borrow only three books. He wants to borrow the Chemistry part-II book only if Chemistry Part-I can also be borrowed. Find the number of ways he can choose three books that he wants to borrow.

Answer the following:

30 objects are to be divided in three groups containing 7, 10, 13 objects. Find the number of distinct ways for doing so.

A student passes an examination if he secures a minimum in each of the 7 subjects. Find the number of ways a student can fail.

Answer the following:

Nine friends decide to go for a picnic in two groups. One group decides to go by car and the other group decides to go by train. Find the number of different ways of doing so if there must be at least 3 friends in each group.

Answer the following:

A hall has 12 lamps and every lamp can be switched on independently. Find the number of ways of illuminating the hall.

Answer the following:

How many quadratic equations can be formed using numbers from 0, 2, 4, 5 as coefficients if a coefficient can be repeated in an equation?

Answer the following:

How many six-digit telephone numbers can be formed if the first two digits are 45 and no digit can appear more than once?

Answer the following:

A question paper has 6 questions. How many ways does a student have to answer if he wants to solve at least one question?

Answer the following:

Find the number of ways of dividing 20 objects in three groups of sizes 8, 7 and 5

Answer the following:

There are 4 doctors and 8 lawyers in a panel. Find the number of ways for selecting a team of 6 if at least one doctor must be in the team

Answer the following:

Four parallel lines intersect another set of five parallel lines. Find the number of distinct parallelograms formed

Answer the following:

There are 12 distinct points A, B, C, ....., L, in order, on a circle. Lines are drawn passing through each pair of points how many lines are there in total.

Answer the following:

There are 12 distinct points A, B, C, ....., L, in order, on a circle. Lines are drawn passing through each pair of points how many lines pass through D.

Answer the following:

There are 12 distinct points A, B, C, ....., L, in order, on a circle. Lines are drawn passing through each pair of points how many triangles are determined by lines.

Answer the following:

There are 12 distinct points A, B, C, ....., L, in order, on a circle. Lines are drawn passing through each pair of points how many triangles have on vertex C.

## Chapter 3: Permutations and Combination

## Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 3 - Permutations and Combination

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 3 (Permutations and Combination) 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 Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 3 Permutations and Combination are Fundamental Principles of Counting, Invariance Principle, Factorial Notation, Permutations When All Objects Are Distinct, Permutations When Repetitions Are Allowed, Permutations When Some Objects Are Identical, Circular Permutations, Properties of Permutations, Properties of Combinations, Combination, Permutations.

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