#### Chapters

## Chapter 6: Permutations and Combinations

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.1 [Pages 72 - 73]

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

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 boy? What is the answer if the monitor must be a girl?

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 has 3 rings and each ring has 5 letters. Determine the maximum number of trials that may be required to open the lock.

In a test that has 5 true/false questions, no student has got all correct answers and no sequence of answers is repeated. What is the maximum number of students for this to be possible?

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 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 3 if digits are not repeated?

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.2 [Pages 74 - 76]

Evaluate: 8!

Evaluate: 6!

Evaluate: 8! – 6!

Evaluate: (8 – 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 × 25

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

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

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

Find n if: `("n"!)/(3!("n" - 5)!) : ("n"!)/(5!("n" - 7)!)` = 10:3

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

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

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!)`

Find the value of: `(8! + 5(4!))/(4! - 12)`

Find the value of: `(5(26!) + (27!))/(4(27!) - 8(26!)`

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

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.3 [Page 81]

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 letters of the word HISTORY can be arranged if Y and T are together.

Find the number of ways letters of the word HISTORY can be arranged if Y is next to T.

Find the number of arrangements of the letters in the word BERMUDA so that consonants and vowels are in the same relative positions.

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 the 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 distinct English letters followed by two non-zero distinct digits. Find the number of such code words.

A code word is formed by two distinct English letters followed by two non-zero distinct digits. 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.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.4 [Pages 82 - 83]

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

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

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

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

You have 2 identical books on English, 3 identical books in 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 M and T 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 letter 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 the two R’s and two A’s are not 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 2?

Find the number of distinct words formed from letters in the word INDIAN. How many of them have 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 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.5 [Page 85]

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

A party has 20 participants and a host. 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 arrangements where two specified delegates are always together.

Delegates from 24 countries participate in a round table discussion. Find the number of seating arrangements 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 20 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 sitting 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 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.6 [Pages 89 - 90]

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 `""^(2"n")"C"_3: ""^"n""C"_2` = 52:3

Find n if `""^"n""C"_("n" - 3)` = 84

Find r if `""^14"C"_(2"r"): ""^10"C"_(2"r" - 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 r.

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, 5 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-shaded polygon. In particular, find the number of diagonals when: n = 10

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

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

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 12 vowels are chosen?

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Exercise 6.7 [Page 90]

Find n if `""^"n""C"_8 = ""^"n""C"_12`

Find n, if `""^23"C"_(3"n") = ""^23"C"_(2"n" + 3)`

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

Find n, if `""^(2"n")"C"_("r" - 1) = ""^(2"n")"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 + ""^17"C"_5`

Find the differences between the largest values in the following: `""^14"C"_r "and" ""^12"C"_r`

Find the differences between the largest values in the following: `""^13"C"_r "and" ""^8"C"_r`

Find the differences between the largest 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 the 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 questions 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 (Commerce) 11th Standard HSC Maharashtra State Board Chapter 6 Permutations and Combinations Miscellaneous Exercise 6 [Pages 92 - 93]

Find the value of `"r" (""^56"C"_("r" + 6)): ""^54"P"_("r" - 1)`= 30800 : 1

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

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

Find the number of ways of distributing n balls in n cells. What will be the number of ways if each cell must be occupied?

Find the number of ways of distributing n balls in n cells. What will be the number of ways if each cell must be occupied?

Thane is the 20^{th} station from C.S.T. If a passenger can purchase a ticket from any station to any other station, how many different ticket must be available at the booking window?

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 letters passwords can be formed using these letters?

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

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.

A student finds 7 books of his interest, but can borrow only three books. He wants to borrow 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.

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/she secures a minimum in each of the 7 subjects. Find the number of ways a student can fail.

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.

Five balls are to be placed in three boxes, where each box can contain up to five balls. Find the number of ways if no box is to remain empty.

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

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

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

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

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

There are 8 doctors and 4 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.

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

## Chapter 6: Permutations and Combinations

## Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board chapter 6 - Permutations and Combinations

Balbharati solutions for Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board chapter 6 (Permutations and Combinations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Commerce) 11th Standard HSC 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 (Commerce) 11th Standard HSC Maharashtra State Board chapter 6 Permutations and Combinations are Introduction of Permutations and Combinations, Fundamental Principles of Counting, Concept of Addition Principle, Concept of Multiplication Principle, Concept of Factorial Function, Permutations When All Objects Are Distinct, Permutations When Repetitions Are Allowed, Permutations When All Objects Are Not Distinct, Circular Permutations, Properties of Permutations, Properties of Combinations, Combination, Permutations.

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