Find the coefficient of x4 in (1 – x)2 (2 + x)5 using binomial theor

1.	Find the coefficient of x4 in (1 – x)2 (2 + x)5 using binomial theorem.
Answer» Given, (1 – x)² (2 + x)⁵ = (1 + x² - 2) (1 + ⁵C₁ 2⁵^{– 1}x¹ + ⁵C₂ 2 ^5–2 x² + ⁵C₃ 2 ^5–3 x³ + ⁵C₄ 2^{5 – 4}x⁴ +⁵C₅ 2^{5 – 5} x⁵) = (1 + x²- 2x) (1 + 5. 2⁴ .x + 10. 2³ x² + 10 . 2² x³ + 5 . 2 . x⁴ + x⁵ ) = (1 + x² – 2x) (1 + 80x + 80x² + 40x³ + 10x⁴ + x 5 ) ∴ Coefficient of x⁴= 80 – 80 + 10 = 10

Find the coefficient of x4 in (1 – x)2 (2 + x)5 using binomial theorem.

Answer»

Given, (1 – x)² (2 + x)⁵

= (1 + x² - 2) (1 + ⁵C₁ 2⁵^{– 1}x¹ + ⁵C₂ 2 ^5–2 x² + ⁵C₃ 2 ^5–3 x³ + ⁵C₄ 2^{5 – 4}x⁴ +⁵C₅ 2^{5 – 5} x⁵)

= (1 + x²- 2x) (1 + 5. 2⁴ .x + 10. 2³ x² + 10 . 2² x³ + 5 . 2 . x⁴ + x⁵ )

= (1 + x² – 2x) (1 + 80x + 80x² + 40x³ + 10x⁴ + x 5 )

∴ Coefficient of x⁴= 80 – 80 + 10 = 10