#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

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

Chapter 8: Transformation formulae

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

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

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

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

## Chapter 17: Combinations

#### 17.1 [Pages 8 - 9]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 17 Combinations 17.1 [Pages 8 - 9]

Evaluate the following:

^{14}*C*_{3}

Evaluate the following:

^{12}*C*_{10}

Evaluate the following:

^{35}*C*_{35}

Evaluate the following:

^{n + }^{1}*C _{n}*

Evaluate the following:

If ^{n}C_{12} = ^{n}C_{5}, find the value of *n*.

If ^{n}C_{4} = ^{n}C_{6}, find ^{12}*C _{n}*.

If ^{n}C_{10} = ^{n}C_{12}, find ^{23}*C _{n}*.

f ^{24}*C _{x}* =

^{24}

*C*

_{2x}

_{ + 3}, find

*x*.

If ^{18}*C _{x}* =

^{18}

*C*

_{x}_{ + 2}, find

*x*.

If ^{15}*C _{3r}* =

^{15}

*C*

_{r}_{ + 3}, find

*r*.

If ^{8}*C _{r}* −

^{7}

*C*

_{3}=

^{7}

*C*

_{2}, find

*r*.

If ^{15}*C _{r}* :

^{15}

*C*

_{r}_{ − 1}= 11 : 5, find

*r*.

If ^{n +}^{2}*C*_{8} : ^{n − }^{2}*P*_{4} = 57 : 16, find *n*.

If ^{28}*C*_{2r} : ^{24}*C*_{2r}_{ − 4} = 225 : 11, find* r*.

If ^{n}C_{4} , ^{n}C_{5} and ^{n}C_{6}_{ }are in A.P., then find *n*.

If ^{2}^{n}C_{3} : ^{n}C_{2} = 44 : 3, find *n*.

If ^{16}*C _{r}* =

^{16}

*C*

_{r}_{ + 2}, find

^{r}C_{4}.

If *α* = ^{m}C_{2}, then find the value of ^{α}C_{2}.

Prove that the product of 2*n* consecutive negative integers is divisible by (2*n*)!

For all positive integers n, show that ^{2}^{n}C_{n} + ^{2}^{n}C_{n}_{ − 1} = `1/2` ^{2n + 2}C_{n+1}

Prove that: ^{4}^{n}C_{2n} : ^{2}* ^{n}C_{n}* = [1 · 3 · 5 ... (4

*n*− 1)] : [1 · 3 · 5 ... (2

*n*− 1)]

^{2}.

Evaluate

Let *r* and *n* be positive integers such that 1 ≤ *r* ≤ *n*. Then prove the following:

Let *r* and *n* be positive integers such that 1 ≤ *r* ≤ *n*. Then prove the following:*n* ·_{ n − }_{1}*C _{r}*

_{ − 1}= (

*n*−

*r*+ 1)

^{n}C_{r}_{ − 1}

Let *r* and *n* be positive integers such that 1 ≤ *r* ≤ *n*. Then prove the following:

Let *r* and *n* be positive integers such that 1 ≤ *r* ≤ *n*. Then prove the following:

* ^{n}C_{r}* + 2 ·

^{n}C_{r}_{ }_{− 1}+

^{n}C_{r}_{ − 2}=

^{n + }

^{2}

*C*.

_{r}#### 17.2 [Pages 15 - 17]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 17 Combinations 17.2 [Pages 15 - 17]

From a group of 15 cricket players, a team of 11 players is to be chosen. In how many ways can this be done?

How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls?

In how many ways can a student choose 5 courses out of 9 courses if 2 courses are compulsory for every student?

In how many ways can a football team of 11 players be selected from 16 players? How many of these will

include 2 particular players?

In how many ways can a football team of 11 players be selected from 16 players? How many of these will

exclude 2 particular players?

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

a particular professor is included.

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

a particular student is included.

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

a particular student is excluded.

How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?

From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition; at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?

How many different selections of 4 books can be made from 10 different books, if

there is no restriction;

How many different selections of 4 books can be made from 10 different books, if

two particular books are always selected;

How many different selections of 4 books can be made from 10 different books, if two particular books are never selected?

From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer

From 4 officers and 8 jawans in how many ways can 6 be chosen. to include at least one officer?

A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the teams be constituted?

A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?

In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make the choice.

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?

There are 10 points in a plane of which 4 are collinear. How many different straight lines can be drawn by joining these points.

Find the number of diagonals of , 1.a hexagon

Find the number of diagonals of (ii) a polygon of 16 sides.

How many triangles can be obtained by joining 12 points, five of which are collinear?

In how many ways can a committee of 5 persons be formed out of 6 men and 4 women when at least one woman has to be necessarily selected?

In a village, there are 87 families of which 52 families have at most 2 children. In a rural development programme, 20 families are to be helped chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made?

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl?

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (ii) at least one boy and one girl?

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has(iii) at least 3 girls?

A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?

Find the number of (i) diagonals

Find the number of (ii) triangles

Determine the number of 5 cards combinations out of a deck of 52 cards if at least one of the 5 cards has to be a king?

We wish to select 6 persons from 8, but if the person *A* is chosen, then *B* must be chosen. In how many ways can the selection be made?

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Determine the number of 5 cards combinations out of a deck of 52 cards if there is exactly one ace in each combination.

In how many ways can one select a cricket team of eleven from 17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers?

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:exactly 3 girls?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: at least 3 girls?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:at most 3 girls?

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

A parallelogram is cut by two sets of m lines parallel to its sides. Find the number of parallelograms thus formed.

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (ii) triangles can be formed by joining them?

#### 17.3 [Page 23]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 17 Combinations 17.3 [Page 23]

How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?

There are 10 persons named\[P_1 , P_2 , P_3 , . . . . , P_{10}\]

Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement *P*_{1} must occur whereas *P*_{4} and *P*_{5} do not occur. Find the number of such possible arrangements.

How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if (i) 4 letters are used at a time

How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used at a time

How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used but first letter is a vowel?

Find the number of permutations of *n* distinct things taken *r *together, in which 3 particular things must occur together.

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?

Find the number of permutations of *n* different things taken *r* at a time such that two specified things occur together?

Find the number of ways in which : (a) a selection

Find the number of ways in which : (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'.

How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?

A business man hosts a dinner to 21 guests. He is having 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests?

Find the number of combinations and permutations of 4 letters taken from the word 'EXAMINATION'.

A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated?

#### [Page 24]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 17 Combinations [Page 24]

Write \[\sum^m_{r = 0} \ ^{n + r}{}{C}_r\] in the simplified form.

If ^{35}*C _{n}*

_{ +7}=

^{35}

*C*

_{4n}

_{ − 2}, then write the values of

*n*.

Write the number of diagonals of an n-sided polygon.

Write the expression ^{n}C_{r}_{ +1} + ^{n}C_{r}_{ − 1} + 2 × * ^{n}C_{r} *in the simplest form.

Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]

There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope.

Write the maximum number of points of intersection of 8 straight lines in a plane.

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.

Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.

Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.

#### [Pages 25 - 26]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 17 Combinations [Pages 25 - 26]

If ^{20}*C _{r}* =

^{20}

*C*

_{r−10}, then

^{18}

*C*is equal to

_{r}4896

816

1632

nont of these

If ^{20}*C _{r}* =

^{20}

*C*

_{r}_{+ 4}, then

^{r}C_{3}is equal to

54

56

58

none of these

If ^{15}*C*_{3r} = ^{15}*C _{r}*

_{ + 3}, then

*r*is equal to

5

4

3

2

If ^{20}*C _{r}*

_{ + 1}=

^{20}

*C*

_{r}_{ − 1}, then

*r*is equal to

10

11

19

12

If *C* (*n*, 12) = *C* (*n*, 8), then *C* (22, *n*) is equal to

231

210

252

303

If ^{m}C_{1}_{ }= ^{n}C_{2} , then

2

*m*=*n*2

*m*=*n*(*n*+ 1)2

*m*=*n*(*n*− 1)2

*n*=*m*(*m*− 1)

If ^{n}C_{12} = ^{n}C_{8} , then *n* =

20

12

6

30

If * ^{n}C_{r}* +

^{n}C_{r}_{ + 1}=

^{n + }

^{1}

*C*, then

_{x}*x*=

r

*r*− 1n

*r*+ 1

If\[\ ^{( a^2 - a)}{}{C}_2 = \ ^{( a^2 - a)}{}{C}_4\] , then *a* =

2

3

4

none of these

^{5}*C*_{1} + ^{5}*C*_{2}_{ }+ ^{5}*C*_{3} + ^{5}*C*_{4} +^{5}*C*_{5} is equal to

30

31

32

33

Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to

60

120

7200

none of these

There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of them are collinear, is

62

63

64

65

Three persons enter a railway compartment. If there are 5 seats vacant, in how many ways can they take these seats?

60

20

15

125

In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one women?

246

222

186

none of these

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is

45

40

39

38

There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of eleven be selected from them so as to include at least two bowlers?

72

78

42

none of these

If *C*_{0} + *C*_{1} + *C*_{2} + ... + *C _{n}* = 256, then

^{2}

^{n}C_{2}is equal to

56

120

28

91

The number of ways in which a host lady can invite for a party of 8 out of 12 people of whom two do not want to attend the party together is

2 ×

^{11}*C*_{7}+^{10}*C*_{8}^{10}*C*_{8}+^{11}*C*_{7}^{12}*C*_{8}−^{10}*C*_{6}none of these

Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given points is

216

156

172

none of these

How many different committees of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve?

(a) 6

(b) 20

(c) 60

(d) 120

6

20

60

120

If ^{43}*C _{r}*

_{ − 6}=

^{43}

*C*

_{3r}

_{ + 1}, then the value of

*r*is

12

8

6

10

14

The number of diagonals that can be drawn by joining the vertices of an octagon is

20

28

8

16

The value of\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + . . . + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\] is

2

^{7}− 12

^{8}− 22

^{8}− 12

^{8}

Among 14 players, 5 are bowlers. In how many ways a team of 11 may be formed with at least 4 bowlers?

265

263

264

275

A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends if two of the friends will not attend the party together is

112

140

164

none of these

If ^{n + }^{1}*C*_{3} = 2 · ^{n}C_{2} , then *n* =

3

4

5

6

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is

6

9

12

18

## Chapter 17: Combinations

## RD Sharma solutions for Class 11 Mathematics Textbook chapter 17 - Combinations

RD Sharma solutions for Class 11 Mathematics Textbook chapter 17 (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 CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 17 Combinations are Concept of Combinations, Fundamental Principle of Counting, Concept of Permutations, Introduction of Permutations and Combinations, Permutation Formula to Rescue and Type of Permutation, Smaller Set from Bigger Set, Derivation of Formulae and Their Connections, Simple Applications of Permutations and Combinations, Factorial N (N!) Permutations and Combinations.

Using RD Sharma Class 11 solutions Combinations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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