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

## Chapter 3: Permutations and Combination

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Miscellaneous Exercise 3
Exercise 3.1 [Pages 47 - 48]

### 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]

Exercise 3.1 | Q 1 | Page 47

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?

Exercise 3.1 | Q 2 | Page 47

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?

Exercise 3.1 | Q 3. (i) | Page 47

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

Exercise 3.1 | Q 3. (ii) | Page 47

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

Exercise 3.1 | Q 4. (i) | Page 47

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

Exercise 3.1 | Q 4. (ii) | Page 47

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

Exercise 3.1 | Q 5 | Page 47

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

Exercise 3.1 | Q 6 | Page 47

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?

Exercise 3.1 | Q 7 | Page 47

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.

Exercise 3.1 | Q 8 | Page 47

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

Exercise 3.1 | Q 9 | Page 47

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

Exercise 3.1 | Q 10 | Page 47

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

Exercise 3.1 | Q 11 | Page 48

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

Exercise 3.1 | Q 12 | Page 48

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

Exercise 3.1 | Q 13 | Page 48

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?

Exercise 3.1 | Q 14 | Page 48

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

Exercise 3.2 [Pages 49 - 50]

### 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]

Exercise 3.2 | Q 1. (i) | Page 49

Evaluate: 8!

Exercise 3.2 | Q 1. (ii) | Page 49

Evaluate: 10!

Exercise 3.2 | Q 1. (iii) | Page 49

Evaluate: 10! – 6!

Exercise 3.2 | Q 1. (iv) | Page 49

Evaluate: (10 – 6)!

Exercise 3.2 | Q 2. (i) | Page 49

Compute: (12!)/(6!)

Exercise 3.2 | Q 2. (ii) | Page 49

Compute: (12/6)!

Exercise 3.2 | Q 2. (iii) | Page 49

Compute: (3 × 2)!

Exercise 3.2 | Q 2. (iv) | Page 49

Compute: 3! × 2!

Exercise 3.2 | Q 2. (v) | Page 49

Compute: (9!)/(3!  6!)

Exercise 3.2 | Q 2. (vi) | Page 49

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

Exercise 3.2 | Q 2. (vii) | Page 49

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

Exercise 3.2 | Q 2. (viii) | Page 49

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

Exercise 3.2 | Q 3. (i) | Page 49

Write in terms of factorial.

5 × 6 × 7 × 8 × 9 × 10

Exercise 3.2 | Q 3. (ii) | Page 49

Write in terms of factorial.

3 × 6 × 9 × 12 × 15

Exercise 3.2 | Q 3. (iii) | Page 49

Write in terms of factorial.

6 × 7 × 8 × 9

Exercise 3.2 | Q 3. (iv) | Page 49

Write in terms of factorial.

5 × 10 × 15 × 20

Exercise 3.2 | Q 4. (i) | Page 49

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

Exercise 3.2 | Q 4. (ii) | Page 49

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

Exercise 3.2 | Q 4. (iii) | Page 49

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

Exercise 3.2 | Q 4. (iv) | Page 49

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

Exercise 3.2 | Q 5. (i) | Page 49

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

Exercise 3.2 | Q 5. (ii) | Page 49

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

Exercise 3.2 | Q 5. (iii) | Page 49

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

Exercise 3.2 | Q 5. (iv) | Page 49

Find n, if (n + 1)! = 42 × (n – 1)!

Exercise 3.2 | Q 5. (v) | Page 49

Find n, if (n + 3)! = 110 × (n + 1)!

Exercise 3.2 | Q 6. (i) | Page 49

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

Exercise 3.2 | Q 6. (ii) | Page 49

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

Exercise 3.2 | Q 6. (iii) | Page 49

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

Exercise 3.2 | Q 6. (iv) | Page 50

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

Exercise 3.2 | Q 6. (v) | Page 50

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

Exercise 3.2 | Q 7 | Page 50

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

Exercise 3.2 | Q 8 | Page 50

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

Exercise 3.2 | Q 9 | Page 50

Show that ((2"n")!)/("n"!) = 2n (2n – 1)(2n – 3) ... 5.3.1

Exercise 3.2 | Q 10. (i) | Page 50

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

Exercise 3.2 | Q 10. (ii) | Page 50

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

Exercise 3.2 | Q 10. (iii) | Page 50

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

Exercise 3.2 | Q 10. (iv) | Page 50

Simplify n[n! + (n – 1)!] + n2(n – 1)! + (n + 1)!

Exercise 3.2 | Q 10. (v) | Page 50

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

Exercise 3.2 | Q 10. (vi) | Page 50

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

Exercise 3.2 | Q 10. (vii) | Page 50

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

Exercise 3.2 | Q 10. (viii) | Page 50

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

Exercise 3.3 [Pages 54 - 55]

### 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]

Exercise 3.3 | Q 1 | Page 54

Find n, if nP6 : nP3 = 120 : 1

Exercise 3.3 | Q 2 | Page 54

Find m and n, if (m+n)P2 = 56 and (m-n)P2 = 12

Exercise 3.3 | Q 3 | Page 54

Find r, if 12Pr–2 : 11Pr–1 = 3 : 14

Exercise 3.3 | Q 4 | Page 54

Show that (n + 1) (nPr) = (n – r + 1) [(n+1)Pr]

Exercise 3.3 | Q 5. (a) | Page 55

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

Exercise 3.3 | Q 5. (b) | Page 55

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

Exercise 3.3 | Q 6. (a) | Page 55

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

Exercise 3.3 | Q 6. (b) | Page 55

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

Exercise 3.3 | Q 6. (c) | Page 55

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

Exercise 3.3 | Q 6. (d) | Page 55

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

Exercise 3.3 | Q 7 | Page 55

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.

Exercise 3.3 | Q 8. (a) | Page 55

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

Exercise 3.3 | Q 8. (b) | Page 55

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

Exercise 3.3 | Q 8. (c) | Page 55

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

Exercise 3.3 | Q 8. (d) | Page 55

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

Exercise 3.3 | Q 8. (e) | Page 55

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

Exercise 3.3 | Q 8. (f) | Page 55

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

Exercise 3.3 | Q 9 | Page 55

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

Exercise 3.3 | Q 10. (a) | Page 55

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

Exercise 3.3 | Q 10. (b) | Page 55

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

Exercise 3.3 | Q 11 | Page 55

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?

Exercise 3.3 | Q 12. (a) | Page 55

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

Exercise 3.3 | Q 12. (b) | Page 55

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

Exercise 3.3 | Q 13 | Page 55

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.

Exercise 3.3 | Q 14 | Page 55

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.

Exercise 3.3 | Q 15. (a) | Page 55

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

Exercise 3.3 | Q 15. (b) | Page 55

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

Exercise 3.3 | Q 16. (i) | Page 55

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

Exercise 3.3 | Q 16. (ii) | Page 55

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

Exercise 3.3 | Q 16. (iii) | Page 55

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

Exercise 3.3 | Q 17. (i) | Page 55

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

Exercise 3.3 | Q 17. (ii) | Page 55

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

Exercise 3.3 | Q 17. (iii) | Page 55

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

Exercise 3.3 | Q 17. (iv) | Page 55

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

Exercise 3.4 [Page 57]

### 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]

Exercise 3.4 | Q 1. (i) | Page 57

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

DIVYA

Exercise 3.4 | Q 1. (ii) | Page 57

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

SHANTARAM

Exercise 3.4 | Q 1. (iii) | Page 57

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

REPRESENT

Exercise 3.4 | Q 1. (iv) | Page 57

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

COMBINE

Exercise 3.4 | Q 1. (v) | Page 57

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

BALBHARATI

Exercise 3.4 | Q 2 | Page 57

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

Exercise 3.4 | Q 3. (a) | Page 57

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

Exercise 3.4 | Q 3. (b) | Page 57

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

Exercise 3.4 | Q 4 | Page 57

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?

Exercise 3.4 | Q 5 | Page 57

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

Exercise 3.4 | Q 6. (a) | Page 57

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

Exercise 3.4 | Q 6. (b) | Page 57

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

Exercise 3.4 | Q 7 | Page 57

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

Exercise 3.4 | Q 8 | Page 57

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

Exercise 3.4 | Q 9 | Page 57

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

Exercise 3.4 | Q 10 | Page 57

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?

Exercise 3.4 | Q 11 | Page 57

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

Exercise 3.4 | Q 12 | Page 57

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

Exercise 3.4 | Q 13 | Page 57

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

Exercise 3.4 | Q 14 | Page 57

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

Exercise 3.4 | Q 15. (a) | Page 57

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

Exercise 3.4 | Q 15. (b) | Page 57

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

Exercise 3.5 [Page 61]

### 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]

Exercise 3.5 | Q 1 | Page 61

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

Exercise 3.5 | Q 2 | Page 61

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?

Exercise 3.5 | Q 3. (a) | Page 61

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

Exercise 3.5 | Q 3. (b) | Page 61

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

Exercise 3.5 | Q 4 | Page 61

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

Exercise 3.5 | Q 5 | Page 61

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

Exercise 3.5 | Q 6. (a) | Page 61

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

Exercise 3.5 | Q 6. (b) | Page 61

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

Exercise 3.5 | Q 7 | Page 61

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

Exercise 3.5 | Q 8 | Page 61

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

Exercise 3.5 | Q 9 | Page 61

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

Exercise 3.5 | Q 10 | Page 61

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

Exercise 3.6 [Pages 64 - 66]

### 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]

Exercise 3.6 | Q 1. (a) | Page 64

Find the value of 15C4

Exercise 3.6 | Q 1. (b) | Page 64

Find the value of 80C2

Exercise 3.6 | Q 1. (c) | Page 64

Find the value of 15C4 + 15C5

Exercise 3.6 | Q 1. (d) | Page 64

Find the value of 20C1619C16

Exercise 3.6 | Q 2. (a) | Page 64

Find n if 6P2 = n 6C2

Exercise 3.6 | Q 2. (b) | Page 64

Find n if 2nC3 : nC2 = 52 : 3

Exercise 3.6 | Q 2. (c) | Page 64

Find n if nCn–3 = 84

Exercise 3.6 | Q 3 | Page 65

Find r if 14C2r : 10C2r–4 = 143 : 10

Exercise 3.6 | Q 4. (a) | Page 65

Find n and r if nPr = 720 and nCn–r = 120

Exercise 3.6 | Q 4. (b) | Page 65

Find n and r if nCr–1 : nCr : nCr+1 = 20 : 35 : 42

Exercise 3.6 | Q 5 | Page 65

If nPr = 1814400 and nCr = 45, find n+4Cr+3

Exercise 3.6 | Q 6 | Page 65

If nCr–1 = 6435, nCr = 5005, nCr+1 = 3003, find rC5

Exercise 3.6 | Q 7 | Page 65

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

Exercise 3.6 | Q 8 | Page 65

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

Exercise 3.6 | Q 9 | Page 65

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.

Exercise 3.6 | Q 10 | Page 65

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

Exercise 3.6 | Q 11. (a) | Page 65

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

Exercise 3.6 | Q 11. (b) | Page 65

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

Exercise 3.6 | Q 11. (c) | Page 65

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

Exercise 3.6 | Q 11. (d) | Page 65

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

Exercise 3.6 | Q 12 | Page 65

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

Exercise 3.6 | Q 13. (a) | Page 65

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

Exercise 3.6 | Q 13. (b) | Page 65

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

Exercise 3.6 | Q 14. (a) | Page 65

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

Exercise 3.6 | Q 14. (b) | Page 65

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

Exercise 3.6 | Q 15 | Page 65

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

Exercise 3.6 | Q 16. (i) | Page 65

Find n if nC8 = nC12

Exercise 3.6 | Q 16. (ii) | Page 65

Find n if 23C3n = 23C2n+3

Exercise 3.6 | Q 16. (iii) | Page 65

Find n if 21C6n = ""^21"C"_(("n"^2 + 5))

Exercise 3.6 | Q 16. (iv) | Page 65

Find n if 2nCr–1 = 2nCr+1

Exercise 3.6 | Q 16. (v) | Page 65

Find n if nCn–2 = 15

Exercise 3.6 | Q 17 | Page 65

Find x if nPr = x nC

Exercise 3.6 | Q 18 | Page 65

Find r if 11C4 + 11C5 + 12C6 + 13C7 = 14Cr

Exercise 3.6 | Q 19 | Page 65

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

Exercise 3.6 | Q 20. (a) | Page 65

Find the differences between the greatest values in the following:

14Cr and 12Cr

Exercise 3.6 | Q 20. (b) | Page 65

Find the differences between the greatest values in the following:

13Cr and 8Cr

Exercise 3.6 | Q 20. (c) | Page 65

Find the differences between the greatest values in the following:

15Cr and 11Cr

Exercise 3.6 | Q 21 | Page 65

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

Exercise 3.6 | Q 22 | Page 65

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?

Exercise 3.6 | Q 23 | Page 65

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?

Exercise 3.6 | Q 24 | Page 66

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?

Exercise 3.6 | Q 25 | Page 66

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?

Exercise 3.6 | Q 26. (a) | Page 66

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

Exercise 3.6 | Q 26. (b) | Page 66

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

Miscellaneous Exercise 3 [Page 67]

### 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]

Miscellaneous Exercise 3 | Q I. (1) | 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

Miscellaneous Exercise 3 | Q I. (2) | Page 67

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

Miscellaneous Exercise 3 | Q I. (3) | Page 67

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!

Miscellaneous Exercise 3 | Q I. (4) | Page 67

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!

Miscellaneous Exercise 3 | Q I. (5) | Page 67

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

Miscellaneous Exercise 3 | Q I. (6) | Page 67

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

Miscellaneous Exercise 3 | Q I. (7) | Page 67

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

Miscellaneous Exercise 3 | Q I. (8) | Page 67

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!

Miscellaneous Exercise 3 | Q I. (9) | Page 67

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

Miscellaneous Exercise 3 | Q I. (10) | Page 67

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

Miscellaneous Exercise 3 [Page 68]

### 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]

Miscellaneous Exercise 3 | Q II (1) | Page 68

Find the value of r if 56Pr+6 : 54Pr+3 = 30800 : 1

Miscellaneous Exercise 3 | Q II. (2) | Page 68

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

Miscellaneous Exercise 3 | Q II. (3) | Page 68

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 40th word?

Miscellaneous Exercise 3 | Q II. (4) | Page 68

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?

Miscellaneous Exercise 3 | Q II. (5) | Page 68

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

Miscellaneous Exercise 3 | Q II. (6) | Page 68

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

Miscellaneous Exercise 3 | Q II. (7) | Page 68

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.

Miscellaneous Exercise 3 | Q II. (8) | Page 68

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

Miscellaneous Exercise 3 | Q II. (9) | Page 68

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.

Miscellaneous Exercise 3 | Q II. (10) | Page 68

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.

Miscellaneous Exercise 3 | Q II. (11) | Page 68

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

Miscellaneous Exercise 3 | Q II. (12) | Page 68

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?

Miscellaneous Exercise 3 | Q II. (13) | Page 68

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

Miscellaneous Exercise 3 | Q II. (14) | Page 68

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

Miscellaneous Exercise 3 | Q II. (15) | Page 68

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

Miscellaneous Exercise 3 | Q II. (16) | Page 68

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

Miscellaneous Exercise 3 | Q II. (17) | Page 68

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

Miscellaneous Exercise 3 | Q II. (18) (i) | Page 68

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.

Miscellaneous Exercise 3 | Q II. (18) (ii) | Page 68

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.

Miscellaneous Exercise 3 | Q II. (18) (iii) | Page 68

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.

Miscellaneous Exercise 3 | Q II. (18) (iv) | Page 68

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

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Miscellaneous Exercise 3

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

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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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