Class 11 Maths Chapter 6 Permutations and Combinations – Exercise 6.4 NCERT Solutions

Introduction

Exercise 6.4 focuses on binomial coefficients and their properties. Students learn how to expand binomial expressions, apply identities involving combinations, and solve problems using properties of ${}^n{C}_r$. This exercise is essential for mastering the Binomial Theorem and probability applications.

Key Formulas

1. Binomial Coefficient:

$${}^n{C}_r=\frac{n!}{r!(n-r)!}$$

2. Symmetry Property:

$${}^n{C}_r{=}^n{C}_{n-r}$$

3. Pascal’s Identity:

$${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$$

4. Binomial Expansion:

$$(x+y{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^{n-r}{y}^r$$

Common Mistakes

• Confusing permutation formula with combination formula.

• Forgetting factorial simplifications.

• Misapplying Pascal’s identity.

• Ignoring symmetry property in simplifications.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Evaluate ${}^8{C}_3$.

$${}^8{C}_3=\frac{8!}{3!\cdot 5!}=\frac{8\cdot 7\cdot 6}{6}=56$$

Q2. Show that ${}^9{C}_4{=}^9{C}_5$.

$${}^9{C}_4=\frac{9!}{4!\cdot 5!}=126,{}^9{C}_5=\frac{9!}{5!\cdot 4!}=126$$

Hence proved.

Q3. Verify Pascal’s identity for $n=5,r=3$.

$${}^5{C}_3{+}^5{C}_2=10+10=20,{}^6{C}_3=20$$

Identity holds.

Q4. Expand $(x+y{)}^4$.

$$(x+y{)}^4{=}^4{C}_0{x}^4{+}^4{C}_1{x}^{3}y{+}^4{C}_2{x}^2{y}^2{+}^4{C}_{3}x{y}^3{+}^4{C}_4{y}^4$$

$$=x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+y^4$$

Q5. Expand $(a+b{)}^5$.

$$(a+b{)}^5=a^5+5a^{4}b+10a^3{b}^2+10a^2{b}^3+5a{b}^4+b^5$$

Q6. Find coefficient of $x^3{y}^2$ in $(x+y{)}^5$.

$$\text{Coefficient}{=}^5{C}_2=10$$

Q7. Find coefficient of $x^2{y}^3$ in $(x+y{)}^5$.

$$\text{Coefficient}{=}^5{C}_3=10$$

Q8. Find coefficient of $x^4{y}^2$ in $(x+y{)}^6$.

$$\text{Coefficient}{=}^6{C}_2=15$$

Q9. Find coefficient of $x^2{y}^4$ in $(x+y{)}^6$.

$$\text{Coefficient}{=}^6{C}_4=15$$

Q10. Expand $(x+y{)}^3$.

$$(x+y{)}^3=x^3+3x^{2}y+3x{y}^2+y^3$$

FAQs (10)

FAQ1. What is binomial coefficient? ${}^n{C}_r=\frac{n!}{r!(n-r)!}$.

FAQ2. What is Pascal’s identity? ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$.

FAQ3. What is binomial expansion? $(x+y{)}^n={\sum }^n{C}_r{x}^{n-r}{y}^r$.

FAQ4. What is symmetry property? ${}^n{C}_r{=}^n{C}_{n-r}$.

FAQ5. What is coefficient in binomial expansion? The binomial coefficient ${}^n{C}_r$.

FAQ6. What is value of ${}^n{C}_0$? Always 1.

FAQ7. What is value of ${}^n{C}_n$? Always 1.

FAQ8. Why is Exercise 6.4 important? It builds foundation for binomial theorem.

FAQ9. Can binomial coefficients be negative? No, they are always non‑negative integers.

FAQ10. What is practical use of binomial coefficients? Used in probability, algebra, and combinatorics.

Conclusion

Exercise 6.4 covers binomial coefficients and expansions with solved examples and FAQs. Mastering these problems helps students in binomial theorem, probability, and advanced algebra.

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New Syllabus-Class 11 Maths Chapter 6 Permutations and Combinations – Exercise 6.4 NCERT Solutions