# RD Sharma solutions for Class 11 Mathematics Textbook chapter 17 - Combinations [Latest edition]

## Chapter 17: Combinations

Exercise 17.1Exercise 17.2Exercise 17.3Others
Exercise 17.1[Pages 8 - 9]

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

Exercise 17.1 | Q 1.1 | Page 8

Evaluate the following:

14C3

Exercise 17.1 | Q 1.2 | Page 8

Evaluate the following:

12C10

Exercise 17.1 | Q 1.3 | Page 8

Evaluate the following:

35C35

Exercise 17.1 | Q 1.4 | Page 8

Evaluate the following:

n + 1Cn

Exercise 17.1 | Q 1.5 | Page 8

Evaluate the following:

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

Exercise 17.1 | Q 2 | Page 8

If nC12 = nC5, find the value of n.

Exercise 17.1 | Q 3 | Page 8

If nC4 = nC6, find 12Cn.

Exercise 17.1 | Q 4 | Page 8

If nC10 = nC12, find 23Cn.

Exercise 17.1 | Q 5 | Page 8

24Cx = 24C2x + 3, find x.

Exercise 17.1 | Q 6 | Page 8

If 18Cx = 18Cx + 2, find x.

Exercise 17.1 | Q 7 | Page 8

If 15C3r = 15Cr + 3, find r.

Exercise 17.1 | Q 8 | Page 8

If 8Cr − 7C3 = 7C2, find r.

Exercise 17.1 | Q 9 | Page 8

If 15Cr : 15Cr − 1 = 11 : 5, find r.

Exercise 17.1 | Q 10 | Page 8

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

Exercise 17.1 | Q 11 | Page 8

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

Exercise 17.1 | Q 12 | Page 8

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

Exercise 17.1 | Q 13 | Page 8

If 2nC3 : nC2 = 44 : 3, find n.

Exercise 17.1 | Q 14 | Page 8

If 16Cr = 16Cr + 2, find rC4.

Exercise 17.1 | Q 15 | Page 8

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

Exercise 17.1 | Q 16 | Page 8

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

Exercise 17.1 | Q 17 | Page 8

For all positive integers n, show that 2nCn + 2nCn − 1 = 1/2 2n + 2Cn+1

Exercise 17.1 | Q 18 | Page 8

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

Exercise 17.1 | Q 19 | Page 8

Evaluate

$^ {20}{}{C}_5 + \sum^5_{r = 2} {}^{25 - r} C_4$
Exercise 17.1 | Q 20.1 | Page 9

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

$\frac{^{n}{}{C}_r}{^{n}{}{C}_{r - 1}} = \frac{n - r + 1}{r}$
Exercise 17.1 | 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

Exercise 17.1 | Q 20.3 | Page 9

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

$\frac{^{n}{}{C}_r}{^{n - 1}{}{C}_{r - 1}} = \frac{n}{r}$
Exercise 17.1 | 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.

Exercise 17.2[Pages 15 - 17]

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

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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.

Exercise 17.2 | 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.

Exercise 17.2 | 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.

Exercise 17.2 | 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)?

Exercise 17.2 | 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?

Exercise 17.2 | Q 8.1 | Page 16

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

Exercise 17.2 | 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;

Exercise 17.2 | 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?

Exercise 17.2 | 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

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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.

Exercise 17.2 | 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?

Exercise 17.2 | 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.

Exercise 17.2 | Q 15.1 | Page 16

Find the number of diagonals of , 1.a hexagon

Exercise 17.2 | Q 15.2 | Page 16

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

Exercise 17.2 | Q 16 | Page 16

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

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | Q 21.1 | Page 16

Find the number of (i) diagonals

Exercise 17.2 | Q 21.2 | Page 16

Find the number of (ii) triangles

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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.

Exercise 17.2 | 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.

Exercise 17.2 | 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?

Exercise 17.2 | 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.

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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?

Exercise 17.2 | 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.

Exercise 17.2 | 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

Exercise 17.2 | 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?

Exercise 17.3[Page 23]

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

Exercise 17.3 | Q 1 | Page 23

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

Exercise 17.3 | 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.

Exercise 17.3 | 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

Exercise 17.3 | 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

Exercise 17.3 | 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?

Exercise 17.3 | Q 4 | Page 23

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

Exercise 17.3 | Q 5 | Page 23

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

Exercise 17.3 | 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?

Exercise 17.3 | Q 7.1 | Page 23

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

Exercise 17.3 | 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'.

Exercise 17.3 | Q 8 | Page 23

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

Exercise 17.3 | 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?

Exercise 17.3 | Q 10 | Page 23

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

Exercise 17.3 | 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?

[Page 24]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 17 Combinations[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.

[Pages 25 - 26]

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

Exercise 17.1Exercise 17.2Exercise 17.3Others

## 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.

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