The coefficient of ^ 99 in the expansion of (x-1)(x-2)(x-3) . ....

Coefficient of $x^1$ in \left[(x-1)(x-2)\right. or \left.\left(x^2-3 x+2\right)\right]=-3=-1-2=-(1+2)$Coefficient of$x^2$in \left[(x-1)(x-2)(x-3)\right.$ or \left(x^3-6 x^2+5 x\right.$$\begin{aligned} & -6)] \\ & =-6=-[1+2+3] \end{aligned}$$$Coefficient of$x^3$in$[(x-1)(x-2)(x-3)(x-4)$or$$\begin{aligned} & \left.\left(\mathrm{x}^4-10 \mathrm{x}^3-29 \mathrm{x}^2-11 \mathrm{x}+24\right)\right] \\ & =-10=-[1+2+3+4] \end{aligned}$$\therefore$ Coefficient of $\mathrmx^{99}$ in $[(\mathrm{x}-1)(\mathrm{x}-2) \ldots \ldots \ldots .(\mathrm{x}-100)]=-[1+2+3+\ldots . .+100]=\frac{-100(100+1)}{2}=-5050 .