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

## Chapter 17: Combinations

Ex. 17.10Ex. 17.20Ex. 17.30Others

#### Chapter 17: Combinations Exercise 17.10 solutions [Pages 8 - 9]

Ex. 17.10 | Q 1.1 | Page 8

Evaluate the following:

14C3

Ex. 17.10 | Q 1.2 | Page 8

Evaluate the following:

12C10

Ex. 17.10 | Q 1.3 | Page 8

Evaluate the following:

35C35

Ex. 17.10 | Q 1.4 | Page 8

Evaluate the following:

n + 1Cn

Ex. 17.10 | Q 1.5 | Page 8

Evaluate the following:

$\sum^5_{r = 1} {}^5 C_r$

Ex. 17.10 | Q 2 | Page 8

If nC12 = nC5, find the value of n.

Ex. 17.10 | Q 3 | Page 8

If nC4 = nC6, find 12Cn.

Ex. 17.10 | Q 4 | Page 8

If nC10 = nC12, find 23Cn.

Ex. 17.10 | Q 5 | Page 8

24Cx = 24C2x + 3, find x.

Ex. 17.10 | Q 6 | Page 8

If 18Cx = 18Cx + 2, find x.

Ex. 17.10 | Q 7 | Page 8

If 15C3r = 15Cr + 3, find r.

Ex. 17.10 | Q 8 | Page 8

If 8Cr − 7C3 = 7C2, find r.

Ex. 17.10 | Q 9 | Page 8

If 15Cr : 15Cr − 1 = 11 : 5, find r.

Ex. 17.10 | Q 10 | Page 8

If n +2C8 : n − 2P4 = 57 : 16, find n.

Ex. 17.10 | Q 11 | Page 8

If 28C2r : 24C2r − 4 = 225 : 11, find r.

Ex. 17.10 | Q 12 | Page 8

If nC4 , nC5 and nC6 are in A.P., then find n.

Ex. 17.10 | Q 13 | Page 8

If 2nC3 : nC2 = 44 : 3, find n.

Ex. 17.10 | Q 14 | Page 8

If 16Cr = 16Cr + 2, find rC4.

Ex. 17.10 | Q 15 | Page 8

If α = mC2, then find the value of αC2.

Ex. 17.10 | Q 16 | Page 8

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

Ex. 17.10 | Q 17 | Page 8

For all positive integers n, show that 2nCn + 2nCn − 1 =$\frac{1}{2}$

Ex. 17.10 | Q 18 | Page 8

Prove that: 4nC2n : 2nCn = [1 · 3 · 5 ... (4n − 1)] : [1 · 3 · 5 ... (2n − 1)]2.

Ex. 17.10 | Q 19 | Page 8

Evaluate

$^ {20}{}{C}_5 + \sum^5_{r = 2} {}^{25 - r} C_4$
Ex. 17.10 | Q 20.1 | Page 9

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

$\frac{^{n}{}{C}_r}{^{n}{}{C}_{r - 1}} = \frac{n - r + 1}{r}$
Ex. 17.10 | Q 20.2 | Page 9

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
n · n − 1Cr − 1 = (n − r + 1) nCr − 1

Ex. 17.10 | Q 20.3 | Page 9

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

$\frac{^{n}{}{C}_r}{^{n - 1}{}{C}_{r - 1}} = \frac{n}{r}$
Ex. 17.10 | Q 20.4 | Page 9

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

nCr + 2 · nCr − 1 + nCr − 2 = n + 2Cr.

#### Chapter 17: Combinations Exercise 17.20 solutions [Pages 15 - 17]

Ex. 17.20 | Q 1 | Page 15

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

Ex. 17.20 | Q 2 | Page 15

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

Ex. 17.20 | Q 3 | Page 15

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

Ex. 17.20 | Q 4.1 | Page 15

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?

Ex. 17.20 | Q 4.2 | Page 15

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?

Ex. 17.20 | Q 5.1 | Page 15

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.

Ex. 17.20 | Q 5.2 | Page 15

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.

Ex. 17.20 | Q 5.3 | Page 15

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.

Ex. 17.20 | Q 6 | Page 15

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

Ex. 17.20 | Q 7 | Page 16

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?

Ex. 17.20 | Q 8.1 | Page 16

How many different selections of 4 books can be made from 10 different books, if
there is no restriction;

Ex. 17.20 | Q 8.2 | Page 16

How many different selections of 4 books can be made from 10 different books, if
two particular books are always selected;

Ex. 17.20 | Q 8.3 | Page 16

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

Ex. 17.20 | Q 9.1 | Page 16

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

Ex. 17.20 | Q 9.2 | Page 16

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

Ex. 17.20 | Q 10 | Page 16

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?

Ex. 17.20 | Q 11 | Page 16

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?

Ex. 17.20 | Q 12 | Page 16

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.

Ex. 17.20 | Q 13 | Page 16

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?

Ex. 17.20 | Q 14 | Page 16

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

Ex. 17.20 | Q 15.1 | Page 16

Find the number of diagonals of , 1.a hexagon

Ex. 17.20 | Q 15.2 | Page 16

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

Ex. 17.20 | Q 16 | Page 16

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

Ex. 17.20 | Q 17 | Page 16

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?

Ex. 17.20 | Q 18 | Page 16

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?

Ex. 17.20 | Q 19.1 | Page 16

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?

Ex. 17.20 | Q 19.2 | Page 16

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?

Ex. 17.20 | Q 19.3 | Page 16

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?

Ex. 17.20 | Q 20 | Page 16

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?

Ex. 17.20 | Q 21.1 | Page 16

Find the number of (i) diagonals

Ex. 17.20 | Q 21.2 | Page 16

Find the number of (ii) triangles

Ex. 17.20 | Q 22 | Page 16

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?

Ex. 17.20 | Q 23 | Page 16

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?

Ex. 17.20 | Q 24 | Page 16

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

Ex. 17.20 | Q 25 | Page 16

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.

Ex. 17.20 | Q 26 | Page 16

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

Ex. 17.20 | Q 27 | Page 17

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?

Ex. 17.20 | Q 28 | Page 17

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.

Ex. 17.20 | Q 29 | Page 17

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?

Ex. 17.20 | Q 30.1 | Page 17

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?

Ex. 17.20 | Q 30.2 | Page 17

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?

Ex. 17.20 | Q 30.3 | Page 17

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?

Ex. 17.20 | Q 31 | Page 17

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?

Ex. 17.20 | Q 32 | Page 17

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

Ex. 17.20 | Q 33.1 | Page 17

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

Ex. 17.20 | Q 33.2 | Page 17

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?

#### Chapter 17: Combinations Exercise 17.30 solutions [Page 23]

Ex. 17.30 | Q 1 | Page 23

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

Ex. 17.30 | Q 2 | Page 23

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 P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.

Ex. 17.30 | Q 3.1 | Page 23

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

Ex. 17.30 | Q 3.2 | Page 23

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

Ex. 17.30 | Q 3.3 | Page 23

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?

Ex. 17.30 | Q 4 | Page 23

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

Ex. 17.30 | Q 5 | Page 23

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

Ex. 17.30 | Q 6 | Page 23

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

Ex. 17.30 | Q 7.1 | Page 23

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

Ex. 17.30 | Q 7.2 | Page 23

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

Ex. 17.30 | Q 8 | Page 23

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

Ex. 17.30 | Q 9 | Page 23

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?

Ex. 17.30 | Q 10 | Page 23

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

Ex. 17.30 | Q 11 | Page 23

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?

#### Chapter 17: Combinations solutions [Page 24]

Q 1 | Page 24

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

Q 2 | Page 24

If 35Cn +7 = 35C4n − 2 , then write the values of n.

Q 3 | Page 24

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

Q 4 | Page 24

Write the expression nCr +1 + nCr − 1 + 2 × nCr in the simplest form.

Q 5 | Page 24

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

Q 6 | Page 24

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

Q 7 | Page 24

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

Q 8 | Page 24

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

Q 9 | Page 24

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.

Q 10 | Page 24

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

Q 11 | Page 24

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

#### Chapter 17: Combinations solutions [Pages 25 - 26]

Q 1 | Page 25

If 20Cr = 20Cr−10, then 18Cr is equal to

• 4896

• 816

• 1632

•  nont of these

Q 2 | Page 25

If 20Cr = 20Cr + 4 , then rC3 is equal to

• 54

•  56

•  58

• none of these

Q 3 | Page 25

If 15C3r = 15Cr + 3 , then r is equal to

• 5

•  4

• 3

• 2

Q 4 | Page 25

If 20Cr + 1 = 20Cr − 1 , then r is equal to

• 10

• 11

•  19

• 12

Q 5 | Page 25

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

• 231

• 210

•  252

• 303

Q 6 | Page 25

If mC1 nC2 , then

• 2 m = n

• 2 m = n (n + 1)

•  2 m = n (n − 1)

• 2 n = m (m − 1)

Q 7 | Page 25

If nC12 = nC8 , then n =

• 20

• 12

• 6

• 30

Q 8 | Page 25

If nCr + nCr + 1 = n + 1Cx , then x =

•  r

• r − 1

• n

• r + 1

Q 9 | Page 25

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

• 2

•  3

• 4

• none of these

Q 10 | Page 25

5C1 + 5C2 5C3 + 5C4 +5C5 is equal to

• 30

• 31

• 32

• 33

Q 11 | Page 25

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

Q 12 | Page 25

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

Q 13 | Page 25

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

Q 14 | Page 25

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

Q 15 | Page 26

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

Q 16 | Page 26

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

Q 17 | Page 26

If C0 + C1 + C2 + ... + Cn = 256, then 2nC2 is equal to

• 56

• 120

• 28

• 91

Q 18 | Page 26

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 × 11C7 + 10C8

• 10C8 + 11C7

• 12C8 − 10C6

• none of these

Q 19 | Page 26

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

Q 20 | Page 26

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

Q 21 | Page 26

If 43Cr − 6 = 43C3r + 1 , then the value of r is

• 12

•  8

•  6

•  10

• 14

Q 22 | Page 26

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

•  20

• 28

•  8

• 16

Q 23 | Page 26

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

• 27 − 1

•  28 − 2

•  28 − 1

• 28

Q 24 | Page 26

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

Q 25 | Page 26

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

Q 26 | Page 26

If n + 1C3 = 2 · nC2 , then n =

•  3

•  4

• 5

•  6

Q 27 | Page 26

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

Ex. 17.10Ex. 17.20Ex. 17.30Others

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

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

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.

Get the free view of chapter 17 Combinations Class 11 extra questions for Maths and can use shaalaa.com to keep it handy for your exam preparation

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