Class 11 Maths Chapter 6 Permutations and Combinations – Exercise 6.3 NCERT Solutions

Introduction

Exercise 6.3 focuses on combinations. Students learn how to calculate the number of ways to select items from a group, apply the formula for combinations, and solve problems involving binomial coefficients. This exercise is essential for understanding probability, counting principles, and advanced combinatorics.

Key Formula

The number of ways to choose $r$ objects from $n$ distinct objects is given by:

$${}^n{C}_r=\frac{n!}{r!(n-r)!},0\le r\le n$$

Common Mistakes

• Confusing permutations (${}^n{P}_r$) with combinations (${}^n{C}_r$).

• Forgetting factorial simplifications.

• Misapplying symmetry property: ${}^n{C}_r{=}^n{C}_{n-r}$.

• Ignoring boundary cases like ${}^n{C}_0=1$.

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

Q1. Evaluate ${}^5{C}_2$.

$${}^5{C}_2=\frac{5!}{2!(5-2)!}=\frac{120}{2\cdot 6}=10$$

Q2. Evaluate ${}^7{C}_3$.

$${}^7{C}_3=\frac{7!}{3!\cdot 4!}=\frac{5040}{6\cdot 24}=35$$

Q3. Evaluate ${}^6{C}_4$.

$${}^6{C}_4=\frac{6!}{4!\cdot 2!}=\frac{720}{24\cdot 2}=15$$

Q4. Show that ${}^n{C}_r{=}^n{C}_{n-r}$ for $n=8,r=3$.

$${}^8{C}_3=\frac{8!}{3!\cdot 5!}=56,{}^8{C}_5=\frac{8!}{5!\cdot 3!}=56$$

Hence proved.

Q5. Evaluate ${}^9{C}_0$.

$${}^9{C}_0=1$$

Q6. Evaluate ${}^9{C}_9$.

$${}^9{C}_9=1$$

Q7. Evaluate ${}^{1}0C_1$.

$${}^{1}0C_1=10$$

Q8. Evaluate ${}^{1}2C_2$.

$${}^{1}2C_2=\frac{12!}{2!\cdot 10!}=\frac{12\cdot 11}{2}=66$$

Q9. Evaluate ${}^{1}5C_3$.

$${}^{1}5C_3=\frac{15!}{3!\cdot 12!}=\frac{15\cdot 14\cdot 13}{6}=455$$

Q10. Evaluate ${}^{2}0C_4$.

$${}^{2}0C_4=\frac{20!}{4!\cdot 16!}=\frac{20\cdot 19\cdot 18\cdot 17}{24}=4845$$

FAQs (10)

FAQ1. What is combination? Selection of items without regard to order.

FAQ2. What is difference between permutation and combination? Permutation considers order, combination does not.

FAQ3. What is formula for ${}^n{C}_r$? $\frac{n!}{r!(n-r)!}$.

FAQ4. What is value of ${}^n{C}_0$? Always 1.

FAQ5. What is value of ${}^n{C}_n$? Always 1.

FAQ6. What is symmetry property of combinations? ${}^n{C}_r{=}^n{C}_{n-r}$.

FAQ7. What is maximum value of ${}^n{C}_r$? Occurs at $r=\lfloor n\mathrm{/}2\rfloor$.

FAQ8. Why are combinations important? Used in probability and counting problems.

FAQ9. Can ${}^n{C}_r$ be zero? No, for valid $0\le r\le n$.

FAQ10. Why is Exercise 6.3 important? It builds foundation for binomial theorem and probability.

Conclusion

Exercise 6.3 covers combinations with solved examples and FAQs. Mastering these problems helps students in probability, binomial expansion, and advanced combinatorics.

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New Syllabus-Class 11 Maths Chapter 6 Permutations and Combinations – Exercise 6.3 NCERT Solutions