A plane passes through the point $\left(1,1,1\right)$. If $b,c,a$ are the direction ratios of a normal to the plane where $a,b,c (a<b<c)$ are the prime factors of 2001, then the equation of the plane is

The equation of the plane is $b(x-1)+c(y-1)+a(z-1)=0\dots \left(i\right)$  

Now, $2001=3\times 23\times 29\therefore a<b<c\Rightarrow a=3,b=23\text{and}c=29$

Substituting the vales of $a,b,c \text{in }{\mathrm{Eq}}_{\cdot }\left(\mathrm{i}\right),\text{we obtain }23x+29y+3z=55$ as the equation of the required plane.