What is the coefficient of x5 in the expansion of (1 – 2x + 3x2 –

1.	What is the coefficient of x5 in the expansion of (1 – 2x + 3x2 – 4x3 + ..... ∞) –5. (a) \(\frac{10!}{(5!)^2}\)(b) 5–5 (c) 55 (d) \(\frac{10!}{6!\,4! }\)
Answer» Answer: (a) \(\frac{10!}{(5!)^2}\) (1 – 2x + 3x² – 4x³ + ..... ∞)^–5 = {(1 + x)^–2}^–5 = (1 + x)¹⁰ ^∴Coefficient of x5 = ¹⁰C₅ = \(\frac{10!}{5!\,5!}\) = \(\frac{10!}{(5!)^2}\) (∵ (1 + x)ⁿ = 1 + ⁿC₁ x + ⁿC₂ x² + ⁿC₃ x² + ....)

What is the coefficient of x5 in the expansion of (1 – 2x + 3x2 – 4x3 + ..... ∞) –5. (a) \(\frac{10!}{(5!)^2}\)(b) 5–5 (c) 55 (d) \(\frac{10!}{6!\,4! }\)

Answer»

Answer: (a) \(\frac{10!}{(5!)^2}\)

(1 – 2x + 3x² – 4x³ + ..... ∞)^–5

= {(1 + x)^–2}^–5 = (1 + x)¹⁰

^∴Coefficient of x5 = ¹⁰C₅ = \(\frac{10!}{5!\,5!}\) = \(\frac{10!}{(5!)^2}\)

(∵ (1 + x)ⁿ = 1 + ⁿC₁ x + ⁿC₂ x² + ⁿC₃ x² + ....)