Question

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 1

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 + ....)