# Example 13

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Example 13 Find the coefficient of a4 in the product (1 + 2a)4 (2 – a)5 using binomial theorem. We know that (a + b)n = nC0 anb0 +nC1 an – 1 b1 + …. …. + nCn – 1 a1 bn – 1+ nCn a0 bn Hence (a + b)4 = 4C0 a4 b0 + 4C1 a3 b1 + 4C2 a2 b2 + 4C3 a1b3 + 4C4 a0 b4 = 4!0! 4 − 0! a4 + 4!1!(4 − 1)! a3 b + 4!2! 4 − 2! a2 b2 + 4!3!(4 −3)! a2b2 + 4!4!(4 − 4)! b4 = 4!1 × 4! a4 + 4!3! a3 b + 4!2! × 2! a2 b2 + 4!3! × 1! ab3 + 4!1! 0! b4 = a4 + 4 × 3!3! a3b + 4 ×3 ×2!2 ×2! a2 b2 + 4 × 3!3! ab3 + 4!4! × 1 b4 = a4 + 4a3 b + 6a2 b2 + 4 ab3 + b4 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 b4 Putting a = 1 & b = 2a (1 + 2a)4 = 14 + 4 (1)3 (2a) + 6(1)2 (2a)2 + 4(1) (2a)3 + (2a)4 = 1 + 8a + 6(4a2) +4 (8a3) + 16 a4 = 1 + 8a + 24a2 + 32a3 + 16a4 Also, We know that (a + b)n = nC0 an b0 + nC1 an – 1 b1 + ….……. + nCn – 1 a1 bn – 1 + nCn a0bn (a + b)5 = 5C0 a5 b0 + 5C1 a4 b1 + 5C2 a3 b2 + 5C3 a2b3 + 5C4 a b4 + 5C5 a0 b5 = 5!0! 5 − 0! a5 + 5!1! 5 − 1! a4 b1 + 5!2! 5 − 2! a3 b2 + 5!3! 5 − 3! a2b3 + 5!4! 5 − 4! a b4 + 5!5! 5 −5! b5 × 1 = 5!0! × 5! a5 + 5!1! × 4! a4 b + 5!2! 3! a3 b2 + 5!3! 2! a2b3 + 5!4! 1! a b4 + 5!5! 0! b5 = 5!5! a5 + 5 × 4!4! a4 b + 5 × 4 × 3!2 × 1 ×3! a3b2 + 5 × 4 × 3!2 ×1 ×3! a2b3 + 5 × 4!4! ab4 + 5!5! b5 = a5 + 5a4 b1 + 10 a3 b2 + 10 a2b3 + 5a b4 + b5 ∴ (a + b)5 = a5 + 5a4b + 10 a3b2 + 10a2b3 + 5ab4 + b5 Putting a = 2 & b = – a (2 – a)5 = (2)5 + 5(2)4 (–a) + 10 (2)3 (–a)2 + 10 (2)2 (–a)3 + 5(2) (–a)4 + (–a)5 = 32 – 5a (16) + 10a2 (8) – 10 a3 (4) + 10a4 – a5 = 32 – 80a + 80a2 – 40 a3 + 10a4 – a5 Now (1 + 2a)4 (2 – 9)5 = (1 + 8a + 24a2 + 32a3 + 16a4) (32 – 80a + 80a2 – 40a3 + 10a4 – a5) Coefficient of a4 is possible as follow Therefore, coefficient of a4 is –438 Hence, the coefficient of a4 in the given product is – 438

Class 11

Important Question for exams Class 11

- Chapter 1 Class 11 Sets
- Chapter 2 Class 11 Relations and Functions
- Chapter 3 Class 11 Trigonometric Functions
- Chapter 4 Class 11 Mathematical Induction
- Chapter 5 Class 11 Complex Numbers
- Chapter 6 Class 11 Linear Inequalities
- Chapter 7 Class 11 Permutations and Combinations
- Chapter 8 Class 11 Binomial Theorem
- Chapter 9 Class 11 Sequences and Series
- Chapter 10 Class 11 Straight Lines
- Chapter 11 Class 11 Conic Sections
- Chapter 12 Class 11 Introduction to Three Dimensional Geometry
- Chapter 13 Class 11 Limits and Derivatives
- Chapter 14 Class 11 Mathematical Reasoning
- Chapter 15 Class 11 Statistics
- Chapter 16 Class 11 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.