NCERT Solutions for Exercise 7.4 Class 11 Maths Chapter 7

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NCERT Solutions for Exercise 7.4 Class 11 Maths Chapter 7 - Permutations and Combinations

Edited By Vishal kumar | Updated on Nov 07, 2023 01:27 PM IST

NCERT Solutions for Class 11 Maths Chapter 7 Permutations And Combinations Exercise 7.4- Download Free PDF

NCERT Solutions for Class 11 Maths Chapter 7: Permutations and Combinations Exercise 7.4- In the previous exercises of this chapter, you have already learned about the permutations when objects are distinct, permutations when the objects are not distinct, etc. In the NCERT solutions for Class 11 Maths Chapter 7 Exercise 7.4, you will learn about the combinations. If you have a command on permutations it won't take much effort to get a command on the Class 11 Maths chapter 7 exercise 7.4.

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NCERT Solutions for Class 11 Maths Chapter 7 Permutations And Combinations Exercise 7.4- Download Free PDF
Download the PDF of NCERT Solutions for Class 11 Maths Chapter 7 – Permutations and Combinations Exercise 7.4
Access Permutation And Combinations Class 11 Chapter 7 Exercise: 7.4
Question:1 If $^{n} C_{8} =^{n} C_{2},$ find $^{n} C_{2}$
More About NCERT Solutions for Class 11 Maths Chapter 7 Exercise 7.4:-
Benefits of NCERT Solutions for Class 11 Maths Chapter 7 Exercise 7.14:-
Key Features of NCERT 11th Class Maths Exercise 7.4 Answers
NCERT Solutions of Class 11 Subject Wise
Subject Wise NCERT Exampler Solutions

NCERT Solutions for Exercise 7.4 Class 11 Maths Chapter 7 - Permutations and Combinations

Both topics permutations and combinations are very important concepts useful in other topics such as probability. These concepts are very useful to solve some real-life problems like selecting fruits out of some given fruits, and selecting n best books out of N books for the same subject. This is the new topic added in Class 11 which hasn't been studied in the previous classes, so you may find it difficult to understand. You should try to solve more problems in order to get conceptual clarity. If you are not able to solve NCERT problems at first by yourself, you can go through the Class 11 Maths chapter 7 exercise 7.4 solutions. Check the NCERT solutions link to get NCERT solutions for C lasses 6 to 12 at one place.

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**As per the CBSE Syllabus for 2023-24, this chapter has been renumbered as Chapter 6.

Download the PDF of NCERT Solutions for Class 11 Maths Chapter 7 – Permutations and Combinations Exercise 7.4

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Access Permutation And Combinations Class 11 Chapter 7 Exercise: 7.4

Question:1 If $^{n} C_{8} =^{n} C_{2},$ find $^{n} C_{2}$

Answer:

Given : $^{n} C_{8} =^{n} C_{2},$

We know that $^{n} C_{a} =^{n} C_{b}$ $\Rightarrow a = b o r n = a + b$

$^{n} C_{8} =^{n} C_{2},$

$\Rightarrow n = 8 + 2$

$\Rightarrow n = 10$

$^{n} C_{2}$ $=^{10} C_{2}$

$= \frac{10!}{(10 - 2)! 2!}$

$= \frac{10!}{8! 2!}$

$= \frac{10 \times 9 \times 8!}{8! 2!}$

$= 5 \times 9 = 45$

Thus the answer is 45

Question:2(i) Determine n if

$^{2 n} C_{3} :^{n} C_{3} = 12 : 1$

Answer:

Given that : $(i)^{2 n} C_{3} :^{n} C_{3} = 12 : 1$

$\Rightarrow \frac{^{2 n} C_{3}}{^{n} C_{3}} = \frac{12}{1}$

The ratio can be written as

$\Rightarrow \frac{\frac{2 n!}{(2 n - 3)! 3!}}{\frac{n!}{(n - 3)! 3!}} = \frac{12}{1}$

$\Rightarrow \frac{2 n! (n - 3)!}{(2 n - 3)! n!} = \frac{12}{1}$

$\Rightarrow \frac{2 n \times (2 n - 1) \times (2 n - 2) \times (2 n - 3)! \times (n - 3)!}{(2 n - 3)! \times n \times (n - 1) \times (n - 2) \times (n - 3)!} = \frac{12}{1}$

$\Rightarrow \frac{2 n \times (2 n - 1) \times (2 n - 2)}{n \times (n - 1) \times (n - 2)} = \frac{12}{1}$

$\Rightarrow \frac{2 \times (2 n - 1) \times 2}{(n - 2)} = \frac{12}{1}$

$\Rightarrow \frac{4 \times (2 n - 1)}{(n - 2)} = \frac{12}{1}$

$\Rightarrow \frac{(2 n - 1)}{(n - 2)} = \frac{3}{1}$

$\Rightarrow 2 n - 1 = 3 n - 6$

$\Rightarrow 6 - 1 = 3 n - 2 n$

$\Rightarrow n = 5$

Question:2(ii) Determine n if

$^{2 n} C_{3} :^{n} C_{3} = 11 : 1$

Answer:

Given that : $(i i)^{2 n} C_{3} :^{n} C_{3} = 11 : 1$

$\Rightarrow \frac{^{2 n} C_{3}}{^{n} C_{3}} = \frac{11}{1}$

$\Rightarrow \frac{\frac{2 n!}{(2 n - 3)! 3!}}{\frac{n!}{(n - 3)! 3!}} = \frac{11}{1}$

$\Rightarrow \frac{2 n! (n - 3)!}{(2 n - 3)! n!} = \frac{11}{1}$

$\Rightarrow \frac{2 n \times (2 n - 1) \times (2 n - 2) \times (2 n - 3)! \times (n - 3)!}{(2 n - 3)! \times n \times (n - 1) \times (n - 2) \times (n - 3)!} = \frac{11}{1}$

$\Rightarrow \frac{2 n \times (2 n - 1) \times (2 n - 2)}{n \times (n - 1) \times (n - 2)} = \frac{11}{1}$

$\Rightarrow \frac{2 \times (2 n - 1) \times 2}{(n - 2)} = \frac{11}{1}$

$\Rightarrow \frac{4 \times (2 n - 1)}{(n - 2)} = \frac{11}{1}$

$\Rightarrow 8 n - 4 = 11 n - 22$

$\Rightarrow 22 - 4 = 11 n - 8 n$

$\Rightarrow 3 n = 18$

$\Rightarrow n = 6$

Thus the value of n=6

Question:3 How many chords can be drawn through 21 points on a circle?

Answer:

To draw chords 2 points are required on the circle.

To know the number of chords on the circle , when points on the circle are 21.

Combinations =Number of chords $=^{21} C_{2}$

$= \frac{21!}{(21 - 2)! 2!}$

$= \frac{21!}{19! 2!}$

$= \frac{21 \times 20}{2}$

$= 210$

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

Answer:

A team of 3 boys and 3 girls be selected from 5 boys and 4 girls.

3 boys can be selected from 5 boys in $^{5} C_{3}$ ways.

3 girls can be selected from 4 boys in $^{4} C_{3}$ ways.

Therefore, by the multiplication principle, the number of ways in which a team of 3 boys and 3 girls can be selected $=^{5} C_{3} \times^{4} C_{3}$

$= \frac{5!}{2! 3!} \times \frac{4!}{1! 3!}$

$= 10 \times 4 = 40$

Question:5 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.

Answer:

There are 6 red balls, 5 white balls and 5 blue balls.

9 balls have to be selected in such a way that consists of 3 balls of each colour.

3 balls are selected from 6 red balls in $^{6} C_{3}$ .

3 balls are selected from 5 white balls in $^{5} C_{3}$

3 balls are selected from 5 blue balls in $^{5} C_{3}$ .

Hence, by the multiplication principle, the number of ways of selecting 9 balls $=^{6} C_{3} \times^{5} C_{3} \times^{5} C_{3}$

$= \frac{6!}{3! 3!} \times \frac{5!}{2! 3!} \times \frac{5!}{2! 3!}$

$= 20 \times 10 \times 10 = 2000$

Question:6 Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Answer:

In a deck, there is 4 ace out of 52 cards.

A combination of 5 cards is to be selected containing exactly one ace.

Then, one ace can be selected in $^{4} C_{1}$ ways and other 4 cards can be selected in $Double exponent: use braces to clarify$ ways.

Hence, using the multiplication principle, required the number of 5 card combination $Double exponent: use braces to clarify$

$= \frac{4!}{1! 3!} \times \frac{48!}{44! 4!}$

$= 4 \times \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2} = 778320$

Question:7 In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Answer:

Out off, 17 players, 5 are bowlers.

A cricket team of 11 is to be selected such that there are exactly 4 bowlers.

4 bowlers can be selected in $^{5} C_{4}$ ways and 7 players can be selected in $Double exponent: use braces to clarify$ ways.

Thus, using multiplication priciple, number of ways of selecting the team $Double exponent: use braces to clarify$

$= \frac{5!}{1! 4!} \times \frac{12!}{5! 7!}$

$= 5 \times \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2}$

$= 3960$

Question:8 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.

Answer:

A bag contains 5 black and 6 red balls.

2 black balls can be selected in $^{5} C_{2}$ ways and 3 red balls can be selected in $^{6} C_{3}$ ways.

Thus, using multiplication priciple, number of ways of selecting 2 black and 3 red balls $=^{5} C_{2} .^{6} C_{3}$

$= \frac{5!}{2! 3!} \times \frac{6!}{3! 3!}$

$= 10 \times 20 = 200$

Question:9 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?

Answer:

9 courses are available and 2 specific courses are compulsory for every student.

Therefore, every student has to select 3 courses out of the remaining 7 courses.

This can be selected in $^{7} C_{3}$ ways.

Thus, using multiplication priciple, number of ways of selecting courses $=^{7} C_{3}$

$= \frac{7!}{3! 4!}$

$= \frac{7 \times 6 \times 5}{3 \times 2}$

$= 35$

Benefits of NCERT Solutions for Class 11 Maths Chapter 7 Exercise 7.14:-

Exercise 7.4 Class 11 Maths is useful for solving real-life problems also.
Class 11 Maths chapter 7 exercise 7.4 solutions are designed by subject matter experts, so you can rely upon these solutions.
You must have conceptual clarity about the Class 11th Maths chapter 7 exercise 7.4 to understand probability.

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Key Features of NCERT 11th Class Maths Exercise 7.4 Answers

In-Depth Coverage: The 11th class maths exercise 7.4 answers comprehensively address all the exercises problems found in exeercise 7.4 of the NCERT 11th Class Mathematics textbook.
Detailed Explanations: The ex 7.4 class 11 solutions are presented with meticulous explanations, ensuring that students can understand the problem-solving process with clarity.
Precision and Simplicity: The class 11 maths ex 7.4 solution are crafted with precision and written in a simple and straightforward language to make the concepts easily understandable for students.
Correct Mathematical Notation: These class 11 ex 7.4 solutions employ the appropriate mathematical notations and terminology, enabling students to become fluent in mathematical language.
Fostering Conceptual Understanding: The 11th class maths exercise 7.4 answers solutions are designed to deepen students' grasp of mathematical concepts, promoting critical thinking and enhancing their problem-solving skills.
No Cost Access: Typically, these ex 7.4 class 11 answers are accessible at no charge, providing students with the freedom to access and use them whenever needed.
Supplementary Learning Resource: These class 11 ex 7.4 answers can serve as valuable supplementary materials to reinforce classroom teachings and support students in their exam preparation.
Homework and Practice: Students can utilize these answers to cross-check their work, practice solving problems, and improve their overall performance in mathematics.

Also see-

NCERT Solutions of Class 11 Subject Wise

Subject Wise NCERT Exampler Solutions

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Frequently Asked Questions (FAQs)

1. What is the weightage of the probability in JEE Main Maths ?

Generally one question is asked from probability in the Maths portion of JEE Main exam which means it has 3.3% weightage in the JEE Main Maths.

2. Where can I get syllabus for Class 11 Maths ?

Here you will get S yllabus for Class 11 Maths.

3. Can I get NCERT solutions for Class 11 Maths for free ?

You can click here to get NCERT Solutions for Class 11 Maths.

4. Where can I get NCERT exemplar solutions for Class 11 Maths ?

You can click here to get NCERT Exemplar Solutions for Class 11 Maths

5. What is the weightage of the mathematical reasoning in the CBSE Class 11 Maths ?

Mathematical reasoning has 2 marks weightage in the CBSE Class 11 Maths.

6. What is the weighatge of limits in JEE Main Maths ?

Generally one question is asked from limit in the JEE Main Maths which means it has 3.3% weightage in the JEE Main Maths.

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up ? Fat supplies 3.8×10⁷J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms⁻² :

Option 1)

2.45×10⁻³ kg

Option 2)

6.45×10⁻³ kg

Option 3)

9.89×10⁻³ kg

Option 4)

12.89×10⁻³ kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

A particle is projected at 60⁰ to the horizontal with a kinetic energy $K$ . The kinetic energy at the highest point

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

In the reaction,

$2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, $Mg_{3}(PO_{4})_{2}$ will contain 0.25 mole of oxygen atoms?

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol^-1) is

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t²) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m² , the number of rotations made by the pulley before its direction of motion if reversed, is