The coefficient of xk in the expansion of (1-2x-x2)e-x is
(1-k-k2)k!
k2+1k!
1k!
1-kk!
The explanation for the correct option:
Determining the coefficient of xk
We have the given equation as: (1-2x-x2)e-x
using expansion to solve the equation.
(1–2x–x2)e-x=(1–2x–x2)ex=(1–2x–x2)1+x1!+x22!+x33!+…..;∵ex=1+x1!+x22!+x33!+…..
Coefficient ofxk=1k!–2(k-1)!–1(k-2)!
=(1-2k-k(k-1)k!=(1-2k-k2+k)k!=(1-k-k2)k!
Hence option A is correct.